| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To solve the problem, we need to match the given formulae to the missing tags in the solution. Let's go through the solution step by step and identify which formulae fit each <missing X> tag. 1. **<missing 1>**: The solution mentions "Denote the crosspoint of $BS$ and <missing 1> through $H$." This suggests that <missing 1> is a line or a segment that intersects with $BS$ at $H$. Given the context, it seems likely that this is the line $AH$ since $AH$ is mentioned later in the solution and is related to the perpendicular from $A$ to $BC$. Therefore, <missing 1> corresponds to <expression 9>. 2. **<missing 2>**: The solution states "To finishing the solution we need only to prove that <missing 2>." This suggests that <missing 2> is a condition or a statement that needs to be proven to complete the solution. Given the context, it seems likely that this is the statement about the symmetry and the orthocenter. Therefore, <missing 2> corresponds to <expression 5>. 3. **<missing 3>**: The solution mentions "= <missing 3> is incenter of <missing 4>." This suggests that <missing 3> is an angle or a relationship involving angles. Given the context, it seems likely that this is the angle relationship involving $\varphi$. Therefore, <missing 3> corresponds to <expression 3>. 4. **<missing 4>**: The solution states "is incenter of <missing 4>." This suggests that <missing 4> is a triangle. Given the context, it seems likely that this is the triangle $ABC$. Therefore, <missing 4> corresponds to <expression 7>. 5. **<missing 5>**: The solution mentions "Denote <missing 5> is the orthocenter of $\triangle TSS'$." This suggests that <missing 5> is a point. Given the context, it seems likely that this is the point $H$. Therefore, <missing 5> corresponds to <expression 7>. 6. **<missing 6>**: The solution states "<missing 6> points $B, G,$ and $F$ are collinear." This suggests that <missing 6> is a condition or a statement that ensures the collinearity of points. Given the context, it seems likely that this is the statement about the symmetry and the orthocenter. Therefore, <missing 6> corresponds to <expression 5>. 7. **<missing 7>**: The solution mentions "<missing 7> is symmetric to $AF$ with respect <missing 8>." This suggests that <missing 7> is a line or a segment. Given the context, it seems likely that this is the line $AF$. Therefore, <missing 7> corresponds to <expression 13>. 8. **<missing 8>**: The solution states "is symmetric to $AF$ with respect <missing 8>." This suggests that <missing 8> is a point or a line. Given the context, it seems likely that this is the point $H$. Therefore, <missing 8> corresponds to <expression 7>. 9. **<missing 9>**: The solution mentions "Let $H$ be the orthocenter of <missing 9> be the height." This suggests that <missing 9> is a triangle. Given the context, it seems likely that this is the triangle $ABC$. Therefore, <missing 9> corresponds to <expression 7>. 10. **<missing 10>**: The solution states "Let <missing 10> be the circle centered at $H$ with radius $HD.$" This suggests that <missing 10> is a circle. Given the context, it seems likely that this is the circle $\omega$. Therefore, <missing 10> corresponds to <expression 2>. 11. **<missing 11>**: The solution mentions "The <missing 11> meets <missing 12> again at $F' \neq D, HD = HF'$." This suggests that <missing 11> is a circle. Given the context, it seems likely that this is the circle $\Omega$. Therefore, <missing 11> corresponds to <expression 1>. 12. **<missing 12>**: The solution states "meets <missing 12> again at $F' \neq D, HD = HF'$." This suggests that <missing 12> is a line or a segment. Given the context, it seems likely that this is the line $BF'$. Therefore, <missing 12> corresponds to <expression 12>. 13. **<missing 13>**: The solution mentions "Let <missing 13> meets <missing 14> again at $E' \neq F'$." This suggests that <missing 13> is a line or a segment. Given the context, it seems likely that this is the line $E'D$. Therefore, <missing 13> corresponds to <expression 18>. 14. **<missing 14>**: The solution states "meets <missing 14> again at $E' \neq F'$." This suggests that <missing 14> is a circle. Given the context, it seems likely that this is the circle $\Omega$. Therefore, <missing 14> corresponds to <expression 1>. 15. **<missing 15>**: The solution mentions "We use Reim’s theorem for <missing 15> and lines $CDD$ and $BE'F'$." This suggests that <missing 15> is a triangle. Given the context, it seems likely that this is the triangle $ABF$. Therefore, <missing 15> corresponds to <expression 16>. 16. **<missing 16>**: The solution states "and get <missing 16> (this idea was recommended by Leonid Shatunov)." This suggests that <missing 16> is a result or a statement. Given the context, it seems likely that this is the statement about the symmetry and the orthocenter. Therefore, <missing 16> corresponds to <expression 6>. 17. **<missing 17>**: The solution mentions "<missing 17> The point <missing 18> is symmetric to $D$ with respect to <missing 19>." This suggests that <missing 17> is a circle. Given the context, it seems likely that this is the circle $\omega$. Therefore, <missing 17> corresponds to <expression 2>. 18. **<missing 18>**: The solution states "The point <missing 18> is symmetric to $D$ with respect to <missing 19>." This suggests that <missing 18> is a point. Given the context, it seems likely that this is the point $E'$. Therefore, <missing 18> corresponds to <expression 10>. 19. **<missing 19>**: The solution mentions "is symmetric to $D$ with respect to <missing 19>." This suggests that <missing 19> is a line or a segment. Given the context, it seems likely that this is the line $AH$. Therefore, <missing 19> corresponds to <expression 9>. Answer: 9, 5, 3, 7, 7, 5, 13, 7, 7, 2, 1, 12, 18, 1, 16, 6, 2, 10, 9 |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| deepseek-r1 | Answer: 15,5,8,16,19,13,14,15,7,2,2,12,12,11,18,16,17,10,9 |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| gpto1 | **Detailed reasoning:** To solve this problem, we need to match the missing parts in the solution to the given expressions by carefully analyzing the context and the content of each expression. 1. **Missing 1:** The solution says, "Denote the crosspoint of \( BS \) and <missing 1> through \( H \), \(\angle ABS = \varphi.\)" Since \( AS' \) is mentioned earlier as the internal angle bisector, it makes sense that <missing 1> is \( AS' \). This corresponds to **Expression 15: \( AS' \).** 2. **Missing 2:** The solution states, "To finish the solution we need only to prove that <missing 2>." Prior equations derived \( AH^2 = BH \cdot DH \), so to complete the proof, we need to show \( PH = AH \). This matches **Expression 5: \( PH = AH \).** 3. **Missing 3:** In the calculation of \( \angle CBS \), we reach, "\( \angle CBS = \frac{\overset{\Large\frown}{AB}}{2} + \frac{\overset{\Large\frown}{AS}}{2} = <missing 3> \) is incenter of..." Here, earlier in the solution, it's given that \( \overset{\Large\frown}{AS} = 2\varphi \). Therefore, \( \angle CBS = \frac{\overset{\Large\frown}{AB}}{2} + \varphi \). This corresponds to **Expression 8: \( = \angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H \).** 4. **Missing 4:** The phrase "...is incenter of <missing 4>" suggests that a point is the incenter of a triangle. Given the context, it is likely referring to triangle \( ABF \). This matches **Expression 16: \( \triangle ABF \).** 5. **Missing 5:** The solution says, "Denote <missing 5> is the orthocenter of \( \triangle TSS' \)." Considering the expressions, **Expression 19** ends with \( H \) and discusses properties leading to \( H \). So, <missing 5> corresponds to **Expression 19: \( T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H \).** 6. **Missing 6:** After stating certain points are concyclic, we have "<missing 6> points \( B, G, \) and \( F \) are collinear..." The conclusion drawn from previous steps likely leads to **Expression 6: \( AH \implies E' = E \implies F' = F \implies HF = HD. \)** 7. **Missing 7 and 8:** The statement "<missing 7> is symmetric to \( AF \) with respect <missing 8>" suggests that a point is symmetric to \( AF \) with respect to another point or line. This scenario aligns with **Expression 10: \( E' \)** and **Expression 9: \( AH \perp BC \implies AH \perp E'D \implies \).** 8. **Missing 9:** In the lemma, "Let \( H \) be the orthocenter of <missing 9> be the height." It makes sense that \( H \) is the orthocenter of \( \triangle ABC \), which is **Expression 7: \( \triangle ABC, BHD \).** 9. **Missing 10:** "Let <missing 10> be the circle centered at \( H \) with radius \( HD \)." This defines \( \omega \), so <missing 10> is **Expression 17: \( \omega \).** 10. **Missing 11:** "The <missing 11> meets..." Since the circle \( \omega \) is being referenced again, <missing 11> is also **Expression 17: \( \omega \).** 11. **Missing 12:** "...meets <missing 12> again at \( F' \neq D \)." This circle meets \( \Omega \) (the circumcircle), so <missing 12> is **Expression 11: \( \Omega \).** 12. **Missing 13:** "Let <missing 13> meets..." The line \( BF' \) is being discussed, so <missing 13> is **Expression 12: \( BF' \).** 13. **Missing 14:** "...meets <missing 14> again at \( E' \neq F' \)." The line meets \( \Omega \), so <missing 14> is **Expression 11: \( \Omega \).** 14. **Missing 15:** Reim's theorem is used for circles and lines mentioned earlier, which involves \( \omega \) and \( \Omega \), corresponding to **Expression 1: \( \omega, \Omega \).** 15. **Missing 16:** The result derived is about triangle \( ABF \), so <missing 16> is **Expression 16: \( \triangle ABF \).** 16. **Missing 17:** The solution continues with "<missing 17> The point...", referring back to \( \omega \), so <missing 17> is **Expression 17: \( \omega \).** 17. **Missing 18 and 19:** "The point <missing 18> is symmetric to \( D \) with respect to <missing 19>." Here, \( E' \) is symmetric to \( D \) with respect to \( AH \), which are **Expression 10: \( E' \)** and **Expression 9: \( AH \perp BC \implies AH \perp E'D \implies \), respectively. **Answer:** 15, 5, 8, 16, 19, 6, 10, 9, 7, 17, 17, 11, 12, 1, 16, 17, 17, 10, 9 |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| gpt-4o-2024-08-06 | To solve this problem, we need to match the missing parts in the solution with the given expressions. Let's go through the solution step by step and identify which expressions fit the missing parts. 1. **First Missing Part:** - The text mentions "Denote the crosspoint of \( BS \) and <missing 1> through \( H \)." This suggests that we are looking for a geometric object or line that intersects with \( BS \) at point \( H \). - Looking at the expressions, <expression 15> is \( AS' \), which is the internal angle bisector of \( \angle BAC \). This makes sense because \( AS' \) is a significant line in the problem, and it would intersect \( BS \) at some point. 2. **Second Missing Part:** - The text states, "To finishing the solution we need only to prove that <missing 2>." - This is likely a statement or conclusion that needs to be proven. <expression 5> is "PH = AH," which is a statement that could be a key part of the proof. 3. **Third Missing Part:** - The text involves angles and arcs, specifically, "angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\frown} {BS}}{2} = \frac {\overset{\Large\frown} {AB}}{2} + \frac {\overset{\Large\frown} {AS}}{2} = <missing 3>." - This is likely a calculation involving angles. <expression 8> involves angles and seems to fit the context: "= \angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H." 4. **Fourth Missing Part:** - The text mentions "is incenter of <missing 4>." - This suggests a triangle or geometric figure. <expression 16> is "\triangle ABF," which fits the context of being a triangle. 5. **Fifth Missing Part:** - The text states, "Denote <missing 5> is the orthocenter of \triangle TSS'." - This is likely a point or line related to the orthocenter. <expression 19> involves the orthocenter and fits the context: "T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H." 6. **Sixth Missing Part:** - The text mentions "2023 IMO 2 lemma.png <missing 6> points B, G, and F are collinear." - This is likely a statement or conclusion. <expression 6> is "AH \implies E' = E \implies F' = F \implies HF = HD," which seems to be a conclusion or result. 7. **Seventh Missing Part:** - The text states, "<missing 7> is symmetric to AF with respect." - This suggests a point or line related to symmetry. <expression 10> is "E'," which could be symmetric to another point or line. 8. **Eighth Missing Part:** - The text involves symmetry, "<missing 8> We use the lemma and complete the proof." - This is likely a statement or conclusion. <expression 9> is "AH \perp BC \implies AH \perp E'D," which involves perpendicularity and could relate to symmetry. 9. **Ninth Missing Part:** - The text mentions "Let acute triangle \triangle ABC, AB > AC be given. Let H be the orthocenter of <missing 9> be the height." - This is likely a triangle. <expression 7> is "\triangle ABC, BHD," which fits the context of a triangle. 10. **Tenth Missing Part:** - The text states, "Let <missing 10> be the circle centered at H with radius HD." - This is likely a circle or geometric object. <expression 17> is "\omega," which is a circle. 11. **Eleventh Missing Part:** - The text involves a circle, "The <missing 11> meets <missing 12> again at F' \neq D, HD = HF'." - This is likely a circle. <expression 11> is "\Omega," which is a circle. 12. **Twelfth Missing Part:** - The text involves a line, "The <missing 11> meets <missing 12> again at F' \neq D, HD = HF'." - This is likely a line. <expression 12> is "BF'," which is a line. 13. **Thirteenth Missing Part:** - The text states, "Let <missing 13> meets <missing 14> again at E' \neq F'." - This is likely a line or point. <expression 13> is "\implies GF," which could be a line. 14. **Fourteenth Missing Part:** - The text involves a line, "Let <missing 13> meets <missing 14> again at E' \neq F'." - This is likely a line. <expression 14> is "TF," which is a line. 15. **Fifteenth Missing Part:** - The text involves a theorem, "We use Reim’s theorem for <missing 15> and lines CDD and BE'F' and get." - This is likely a geometric object or line. <expression 1> is "\omega, \Omega," which are circles. 16. **Sixteenth Missing Part:** - The text involves a conclusion, "<missing 16> (this idea was recommended by Leonid Shatunov)." - This is likely a statement or conclusion. <expression 18> is "E'D || BC," which is a statement. 17. **Seventeenth Missing Part:** - The text involves a conclusion, "<missing 17> The point." - This is likely a statement or conclusion. <expression 4> is "\omega," which is a circle. 18. **Eighteenth Missing Part:** - The text involves symmetry, "The point <missing 18> is symmetric to D with respect to." - This is likely a point or line. <expression 10> is "E'," which could be symmetric to another point or line. 19. **Nineteenth Missing Part:** - The text involves symmetry, "The point <missing 18> is symmetric to D with respect to <missing 19>." - This is likely a point or line. <expression 19> is "T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H," which involves perpendicularity and could relate to symmetry. Based on this reasoning, the final answer is: Answer: 15, 5, 8, 16, 19, 6, 10, 9, 7, 17, 11, 12, 13, 14, 1, 18, 4, 10, 19 |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| DeepSeek-Coder-V2-Lite-Instruct | <Detailed reasoning> First, let's identify the missing tags in the solution. The solution references several formulae and concepts, and we need to match these with the given expressions. 1. The solution mentions "Denote the crosspoint of $BS$ and <missing 1> through $H$." This suggests that the missing tag corresponds to the line or point where $BS$ intersects something. Given the context, it could be the line $AE$ since $AE$ is perpendicular to $BC$ and $S$ is on the circle $\Omega$. 2. The solution needs to prove that "<missing 2>" which likely refers to a specific line or point. This could be the line tangent to $\omega$ at $P$ meeting $BS$ on the internal angle bisector of $\angle BAC$. 3. The solution states that "To finishing the solution we need only to prove that <missing 2>". This implies that the missing tag in the solution is related to the line tangent to $\omega$ at $P$ meeting $BS$ on the internal angle bisector of $\angle BAC$. 4. The solution mentions "<missing 3>" which likely refers to the circumcircle of $\triangle BDL$. 5. The solution refers to "<missing 4>" which could be the triangle $\triangle BDL$. 6. The solution uses the lemma and completes the proof, mentioning "<missing 6>" which likely refers to the point where $B, G,$ and $F$ are collinear. 7. The solution refers to "<missing 7>" which likely refers to the symmetry of $AF$ with respect to some point. 8. The solution mentions "<missing 8>" which likely refers to the point symmetric to $AF$ with respect to some line. 9. The solution uses "<missing 9>" which likely refers to the triangle $\triangle ABC$ and the height $AH$. 10. The solution mentions "<missing 10>" which likely refers to the point $E'$. 11. The solution uses "<missing 11>" which likely refers to the circle centered at $H$ with radius $HD$. 12. The solution mentions "<missing 12>" which likely refers to the line $BC$. 13. The solution uses "<missing 13>" which likely refers to the line $BE'$. 14. The solution mentions "<missing 14>" which likely refers to the line $AF'$. 15. The solution uses "<missing 15>" which likely refers to the line $AS'$. 16. The solution mentions "<missing 16>" which likely refers to the triangle $\triangle ABF$. 17. The solution uses "<missing 17>" which likely refers to the circle $\omega$. 18. The solution mentions "<missing 18>" which likely refers to the line $E'D$. 19. The solution uses "<missing 19>" which likely refers to the point $T$. Now, let's match these with the given expressions: 1. <missing 1> corresponds to $AE$. 2. <missing 2> corresponds to the line tangent to $\omega$ at $P$ meeting $BS$ on the internal angle bisector of $\angle BAC$. 3. <missing 3> corresponds to the circumcircle of $\triangle BDL$. 4. <missing 4> corresponds to $\triangle BDL$. 5. <missing 5> corresponds to $PH = AH$. 6. <missing 6> corresponds to the point where $B, G,$ and $F$ are collinear. 7. <missing 7> corresponds to the symmetry of $AF$ with respect to some point. 8. <missing 8> corresponds to the point symmetric to $AF$ with respect to some line. 9. <missing 9> corresponds to the triangle $\triangle ABC$ and the height $AH$. 10. <missing 10> corresponds to $E'$. 11. <missing 11> corresponds to the circle centered at $H$ with radius $HD$. 12. <missing 12> corresponds to the line $BC$. 13. <missing 13> corresponds to the line $BE'$. 14. <missing 14> corresponds to the line $AF'$. 15. <missing 15> corresponds to $AS'$. 16. <missing 16> corresponds to the triangle $\triangle ABF$. 17. <missing 17> corresponds to the circle $\omega$. 18. <missing 18> corresponds to the line $E'D$. 19. <missing 19> corresponds to the point $T$. Thus, the expression identifiers that fill the missing parts are: 3, 2, 4, 1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 Answer: 3, 2, 4, 1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| Meta-Llama-3.1-405B-Instruct-Turbo | To solve this problem, we need to carefully read the solution and identify the missing parts. Then, we need to match the formulae to the missing tags in the solution. The first missing part is <missing 1>. The solution mentions that "Denote the crosspoint of $BS$ and <missing 1> through $H$". This suggests that <missing 1> is a line or a curve that intersects with $BS$ at point $H$. Looking at the formulae, we can see that <expression 1> mentions "\omega, \Omega", which are two circles. It's likely that <missing 1> refers to one of these circles. The second missing part is <missing 2>. The solution states that "To finishing the solution we need only to prove that <missing 2>". This suggests that <missing 2> is a statement or a claim that needs to be proven. Looking at the formulae, we can see that <expression 2> simply states "\omega", which is a circle. It's likely that <missing 2> refers to a property or a statement about this circle. The third missing part is <missing 3>. The solution mentions that "\angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\frown} {BS}}{2} = \frac {\overset{\Large\frown} {AB}}{2} + \frac {\overset{\Large\frown} {AS}}{2} = <missing 3>". This suggests that <missing 3> is an angle or an expression related to angles. Looking at the formulae, we can see that <expression 3> mentions "\angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies". This expression seems to match the context of the solution. Continuing this process, we can match the remaining missing parts to the formulae. Here is the detailed reasoning: * <missing 1> corresponds to <expression 1> "\omega, \Omega" * <missing 2> corresponds to <expression 2> "\omega" * <missing 3> corresponds to <expression 3> "\angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies" * <missing 4> corresponds to <expression 4> "\omega" * <missing 5> corresponds to <expression 5> "PH = AH" * <missing 6> corresponds to <expression 6> "AH \implies E' = E \implies F' = F \implies HF = HD" * <missing 7> corresponds to <expression 7> "\triangle ABC, BHD" * <missing 8> corresponds to <expression 8> "= \angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H" * <missing 9> corresponds to <expression 9> "AH \perp BC \implies AH \perp E'D \implies" * <missing 10> corresponds to <expression 10> "E'" * <missing 11> corresponds to <expression 11> "\Omega" * <missing 12> corresponds to <expression 12> "BF'" * <missing 13> corresponds to <expression 13> "\implies GF" * <missing 14> corresponds to <expression 14> "TF" * <missing 15> corresponds to <expression 15> "AS'" * <missing 16> corresponds to <expression 16> "\triangle ABF" * <missing 17> corresponds to <expression 17> "\omega" * <missing 18> corresponds to <expression 18> "E'D || BC" * <missing 19> corresponds to <expression 19> "T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H" Answer: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| Meta-Llama-3.1-70B-Instruct-Turbo | To solve this problem, we need to carefully match the given formulae with the missing tags in the solution. Here's the step-by-step reasoning: 1. The first missing tag is <missing 1>, which is related to the crosspoint of $BS$ and another line. Looking at the formulae, we find <expression 3> which mentions $\angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS'$. This seems to be related to the angles formed by the crosspoint, so we can match <missing 1> with <expression 3>. 2. The second missing tag is <missing 2>, which is a statement about the incenter of a triangle. Looking at the formulae, we find <expression 4> which mentions $\omega$. However, this doesn't seem to be directly related to the incenter. After re-examining the formulae, we find <expression 7> which mentions $\triangle ABC, BHD$. This seems to be related to the incenter, so we can match <missing 2> with <expression 7>. 3. The third missing tag is <missing 3>, which is an expression for an angle. Looking at the formulae, we find <expression 8> which mentions $=\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H$. This seems to be related to the angle expression, so we can match <missing 3> with <expression 8>. 4. The fourth missing tag is <missing 4>, which is related to the incenter of a triangle. Looking at the formulae, we find <expression 4> which mentions $\omega$. This seems to be related to the incenter, so we can match <missing 4> with <expression 4>. 5. The fifth missing tag is <missing 5>, which is related to the orthocenter of a triangle. Looking at the formulae, we find <expression 5> which mentions $PH = AH$. This seems to be related to the orthocenter, so we can match <missing 5> with <expression 5>. 6. The sixth missing tag is <missing 6>, which is a statement about the collinearity of points. Looking at the formulae, we find <expression 6> which mentions $AH \implies E' = E \implies F' = F \implies HF = HD$. This seems to be related to the collinearity, so we can match <missing 6> with <expression 6>. 7. The seventh missing tag is <missing 7>, which is a statement about the symmetry of points. Looking at the formulae, we find <expression 9> which mentions $AH \perp BC \implies AH \perp E'D \implies$. This seems to be related to the symmetry, so we can match <missing 7> with <expression 9>. 8. The eighth missing tag is <missing 8>, which is related to the symmetry of points. Looking at the formulae, we find <expression 10> which mentions $E'$. This seems to be related to the symmetry, so we can match <missing 8> with <expression 10>. 9. The ninth missing tag is <missing 9>, which is related to the orthocenter of a triangle. Looking at the formulae, we find <expression 5> which mentions $PH = AH$. This seems to be related to the orthocenter, so we can match <missing 9> with <expression 5>. 10. The tenth missing tag is <missing 10>, which is related to the circle centered at $H$. Looking at the formulae, we find <expression 11> which mentions $\Omega$. This seems to be related to the circle, so we can match <missing 10> with <expression 11>. 11. The eleventh missing tag is <missing 11>, which is related to the intersection of circles. Looking at the formulae, we find <expression 12> which mentions $BF'$. This seems to be related to the intersection, so we can match <missing 11> with <expression 12>. 12. The twelfth missing tag is <missing 12>, which is related to the intersection of circles. Looking at the formulae, we find <expression 13> which mentions $\implies GF$. This seems to be related to the intersection, so we can match <missing 12> with <expression 13>. 13. The thirteenth missing tag is <missing 13>, which is related to the intersection of circles. Looking at the formulae, we find <expression 14> which mentions $TF$. This seems to be related to the intersection, so we can match <missing 13> with <expression 14>. 14. The fourteenth missing tag is <missing 14>, which is related to the intersection of circles. Looking at the formulae, we find <expression 15> which mentions $AS'$. This seems to be related to the intersection, so we can match <missing 14> with <expression 15>. 15. The fifteenth missing tag is <missing 15>, which is related to the application of Reim's theorem. Looking at the formulae, we find <expression 16> which mentions $\triangle ABF$. This seems to be related to the application of Reim's theorem, so we can match <missing 15> with <expression 16>. 16. The sixteenth missing tag is <missing 16>, which is a statement about the equality of segments. Looking at the formulae, we find <expression 6> which mentions $AH \implies E' = E \implies F' = F \implies HF = HD$. This seems to be related to the equality of segments, so we can match <missing 16> with <expression 6>. 17. The seventeenth missing tag is <missing 17>, which is related to the symmetry of points. Looking at the formulae, we find <expression 18> which mentions $E'D || BC$. This seems to be related to the symmetry, so we can match <missing 17> with <expression 18>. 18. The eighteenth missing tag is <missing 18>, which is related to the symmetry of points. Looking at the formulae, we find <expression 19> which mentions $T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H$. This seems to be related to the symmetry, so we can match <missing 18> with <expression 19>. 19. The nineteenth missing tag is <missing 19>, which is related to the symmetry of points. Looking at the formulae, we find <expression 10> which mentions $E'$. This seems to be related to the symmetry, so we can match <missing 19> with <expression 10>. Answer: 3, 7, 8, 4, 5, 6, 9, 10, 5, 11, 12, 13, 14, 15, 16, 6, 18, 19, 10 |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| Qwen2-72B-Instruct | <Detailed reasoning> In the solution, the first missing part refers to a line or a segment that intersects BS at a point H. This could be any line, but since we're dealing with a geometric problem involving circles and triangles, it's likely referring to a specific line related to the triangle or one of the circles. Given the context, it seems like this could be related to the circumcircle Ω or the circumcircle ω. However, since the next sentence talks about angles related to BS and the fact that AE is perpendicular to BC, it's more likely that this missing part refers to something related to the perpendicular from A to BC, which is mentioned earlier. None of the expressions directly mention this, but since the problem involves proving properties of the configuration, it's reasonable to infer that this missing part might refer to a property or construction related to the given points and lines. Without further context, it's challenging to pinpoint exactly which expression fits here, but given the context of the problem, it seems most likely that this would involve some property of the triangle or the circles involved. Since none of the expressions directly fit this description, I will make an educated guess based on the context provided. The second missing part refers to a statement that needs to be proven to finish the solution. This likely involves showing a relationship between points, lines, or circles that completes the proof. Given the context of the problem, this could involve showing that certain points are collinear, certain lines are parallel, or that certain angles are equal. Expression 3 seems to fit this context as it describes a relationship between angles and points that could potentially complete the proof. The third missing part refers to an angle measure that is equal to the sum of two other angle measures. This likely involves calculating an angle based on known properties of the triangle or the circles. Expression 8 seems to fit this context as it describes an angle measure that is the sum of two other angle measures. The fourth missing part refers to a triangle for which a point is the incenter. This likely involves identifying a triangle for which a given point serves as the incenter. Expression 16 seems to fit this context as it describes a triangle. The fifth missing part refers to a point that is the orthocenter of a triangle. This likely involves identifying the orthocenter of a specific triangle. Expression 19 seems to fit this context as it describes a point that could potentially serve as the orthocenter of a triangle. The sixth missing part refers to a statement about points being collinear. This likely involves showing that three points lie on the same line. Expression 13 seems to fit this context as it describes a line that passes through multiple points. The seventh missing part refers to a line being symmetric to another line with respect to a point. This likely involves showing that a line is symmetric to another line with respect to a specific point. Expression 15 seems to fit this context as it describes a line that could potentially be symmetric to another line with respect to a point. The eighth missing part refers to a point being symmetric to another point with respect to a line. This likely involves identifying a point that is symmetric to another point with respect to a specific line. Expression 18 seems to fit this context as it describes a point that is symmetric to another point with respect to a line. The ninth missing part refers to a triangle for which a point is the orthocenter. This likely involves identifying a triangle for which a given point serves as the orthocenter. Expression 7 seems to fit this context as it describes a triangle for which a point serves as the orthocenter. The tenth missing part refers to a circle centered at a point with a specific radius. This likely involves identifying a circle that is centered at a specific point with a given radius. Expression 9 seems to fit this context as it describes a circle centered at a point with a specific radius. The eleventh missing part refers to a circle meeting another circle again at a point. This likely involves identifying a point where two circles intersect. Expression 11 seems to fit this context as it describes a circle meeting another circle again at a point. The twelfth missing part refers to a line meeting a circle again at a point. This likely involves identifying a point where a line intersects a circle. Expression 12 seems to fit this context as it describes a line meeting a circle again at a point. The thirteenth missing part refers to a line meeting another line again at a point. This likely involves identifying a point where two lines intersect. Expression 14 seems to fit this context as it describes a line meeting another line again at a point. The fourteenth missing part refers to a circle and a line for which a theorem can be applied. This likely involves applying a specific theorem to a circle and a line. Expression 17 seems to fit this context as it describes a circle and a line for which a theorem can be applied. The fifteenth missing part refers to a line that is parallel to another line. This likely involves identifying a line that is parallel to another line. Expression 18 seems to fit this context as it describes a line that is parallel to another line. The sixteenth missing part refers to a point that is symmetric to another point with respect to a line. This likely involves identifying a point that is symmetric to another point with respect to a specific line. Expression 19 seems to fit this context as it describes a point that is symmetric to another point with respect to a line. Answer: 1, 3, 8, 16, 19, 13, 15, 18, 7, 9, 11, 12, 14, 17, 18, 19 |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| baichuan4 | First, let's analyze the context and the structure of the solution to determine which formulae might fit into the missing tags. 1. **<missing 1>**: This is likely referring to a line or a circle that intersects with $BS$. Given the context, it seems like it could be the circumcircle of triangle $BDL$, which is denoted by $\omega$. So, this might correspond to <expression 1>. 2. **<missing 2>**: This is a statement that needs to be proven. It seems to be related to the properties of the circle $\omega$. Given the context, it could be related to the properties of the incenter or the orthocenter of the triangle, which are discussed later in the solution. However, there isn't a direct match with the provided formulae. 3. **<missing 3>**: This is part of an equation that calculates an angle. It seems to be related to the angle bisector and the arcs on the circumcircle. Given the context, it could be related to the angle bisector of $\angle BAC$, which is mentioned earlier. This might correspond to <expression 8>. 4. **<missing 4>**: This is referring to the incenter of a triangle. Given the context, it seems like it could be the incenter of triangle $BDL$, whose circumcircle is $\omega$. So, this might correspond to <expression 4>. 5. **<missing 5>**: This is referring to a point that is the orthocenter of a triangle. Given the context, it seems like it could be the orthocenter of triangle $TSS'$. So, this might correspond to <expression 19>. 6. **<missing 6>**: This is referring to a line that passes through certain points. Given the context, it seems like it could be the line $AH$. So, this might correspond to <expression 6>. 7. **<missing 7>**: This is referring to a line that is symmetric to another line. Given the context, it seems like it could be the line $AF$. However, there isn't a direct match with the provided formulae. 8. **<missing 8>**: This is referring to a line with respect to which another line is symmetric. Given the context, it seems like it could be the line $AH$. So, this might correspond to <expression 9>. 9. **<missing 9>**: This is referring to the orthocenter of a triangle. Given the context, it seems like it could be the orthocenter of triangle $ABC$. So, this might correspond to <expression 7>. 10. **<missing 10>**: This is referring to a circle centered at a point with a certain radius. Given the context, it seems like it could be the circle centered at $H$ with radius $HD$. So, this might correspond to <expression 11>. 11. **<missing 11>**: This is referring to a circle that meets another circle. Given the context, it seems like it could be the circle centered at $H$ with radius $HD$ that meets the circle $\Omega$. So, this might correspond to <expression 12>. 12. **<missing 12>**: This is referring to a line that meets another line. Given the context, it seems like it could be the line $BF'$. So, this might correspond to <expression 13>. 13. **<missing 13>**: This is referring to a line that meets another line. Given the context, it seems like it could be the line $GF$. So, this might correspond to <expression 14>. 14. **<missing 14>**: This is referring to a line that meets another line. Given the context, it seems like it could be the line $TF$. So, this might correspond to <expression 15>. 15. **<missing 15>**: This is referring to a line that meets another line. Given the context, it seems like it could be the line $AS'$. So, this might correspond to <expression 16>. 16. **<missing 16>**: This is referring to a triangle. Given the context, it seems like it could be the triangle $ABF$. So, this might correspond to <expression 17>. 17. **<missing 17>**: This is referring to a circle. Given the context, it seems like it could be the circle $\omega$. So, this might correspond to <expression 18>. 18. **<missing 18>**: This is referring to a line that is parallel to another line. Given the context, it seems like it could be the line $E'D$. So, this might correspond to <expression 19>. 19. **<missing 19>**: This is referring to a point that is symmetric to another point with respect to a line. Given the context, it seems like it could be the point $T$. So, this might correspond to <expression 20>. Based on the above analysis, here is my best guess for the missing formulae: Answer: 1, 8, 4, 19, 6, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19 |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| doubao | <Detailed reasoning>: In the solution, <missing 1> is where the crosspoint of $BS$ and the circumcircle of triangle $BDL$ is mentioned, which is $\omega$. So <missing 1> is filled by <expression 2>. <missing 2> is related to proving something about the incenter, and looking at the expressions, <expression 4> mentions "is incenter of" which seems to fit. For <missing 3>, the expression that shows the relationship with angles is <expression 3>. <missing 4> is related to being the incenter of something, and <expression 4> is a good fit. <missing 5> is the orthocenter, and <expression 5> has $PH = AH$ which is related to the orthocenter. <missing 6> is related to collinearity and <expression 6> mentions $AH$. <missing 7> is related to symmetry with respect to something, and <expression 7> mentions symmetry with respect to $\triangle ABC$. <missing 8> is related to angles, and <expression 8> has the relevant angle expression. <missing 9> is the triangle where the orthocenter is, and <expression 9> has $AH \perp BC$ which is the relevant triangle. <missing 10> is a circle centered at $H$ with radius $HD$, which is <expression 10>. <missing 11> is the circle that meets something else, and <expression 11> is $\Omega$. <missing 12> is the line that the circle meets, and <expression 12> is $BF'$. <missing 13> is related to another line, and <expression 13> is $GF$. <missing 14> is the other line that is met, and <expression 14> is $TF$. <missing 15> is related to Reim's theorem, and <expression 15> is $AS'$. <missing 16> is related to a triangle in the lemma, and <expression 16> is $\triangle ABF$. <missing 17> is the relevant circle again, which is <expression 17>. <missing 18> is related to a symmetry statement, and <expression 18> has the relevant statement. <missing 19> is the point of symmetry, and <expression 19> has the relevant point. Answer: 2, 4, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19> |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| gpto1 | **Detailed reasoning:** To solve this problem, we need to match the missing parts in the solution to the given expressions by carefully analyzing the context and the content of each expression. 1. **Missing 1:** The solution says, "Denote the crosspoint of \( BS \) and <missing 1> through \( H \), \(\angle ABS = \varphi.\)" Since \( AS' \) is mentioned earlier as the internal angle bisector, it makes sense that <missing 1> is \( AS' \). This corresponds to **Expression 15: \( AS' \).** 2. **Missing 2:** The solution states, "To finish the solution we need only to prove that <missing 2>." Prior equations derived \( AH^2 = BH \cdot DH \), so to complete the proof, we need to show \( PH = AH \). This matches **Expression 5: \( PH = AH \).** 3. **Missing 3:** In the calculation of \( \angle CBS \), we reach, "\( \angle CBS = \frac{\overset{\Large\frown}{AB}}{2} + \frac{\overset{\Large\frown}{AS}}{2} = <missing 3> \) is incenter of..." Here, earlier in the solution, it's given that \( \overset{\Large\frown}{AS} = 2\varphi \). Therefore, \( \angle CBS = \frac{\overset{\Large\frown}{AB}}{2} + \varphi \). This corresponds to **Expression 8: \( = \angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H \).** 4. **Missing 4:** The phrase "...is incenter of <missing 4>" suggests that a point is the incenter of a triangle. Given the context, it is likely referring to triangle \( ABF \). This matches **Expression 16: \( \triangle ABF \).** 5. **Missing 5:** The solution says, "Denote <missing 5> is the orthocenter of \( \triangle TSS' \)." Considering the expressions, **Expression 19** ends with \( H \) and discusses properties leading to \( H \). So, <missing 5> corresponds to **Expression 19: \( T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H \).** 6. **Missing 6:** After stating certain points are concyclic, we have "<missing 6> points \( B, G, \) and \( F \) are collinear..." The conclusion drawn from previous steps likely leads to **Expression 6: \( AH \implies E' = E \implies F' = F \implies HF = HD. \)** 7. **Missing 7 and 8:** The statement "<missing 7> is symmetric to \( AF \) with respect <missing 8>" suggests that a point is symmetric to \( AF \) with respect to another point or line. This scenario aligns with **Expression 10: \( E' \)** and **Expression 9: \( AH \perp BC \implies AH \perp E'D \implies \).** 8. **Missing 9:** In the lemma, "Let \( H \) be the orthocenter of <missing 9> be the height." It makes sense that \( H \) is the orthocenter of \( \triangle ABC \), which is **Expression 7: \( \triangle ABC, BHD \).** 9. **Missing 10:** "Let <missing 10> be the circle centered at \( H \) with radius \( HD \)." This defines \( \omega \), so <missing 10> is **Expression 17: \( \omega \).** 10. **Missing 11:** "The <missing 11> meets..." Since the circle \( \omega \) is being referenced again, <missing 11> is also **Expression 17: \( \omega \).** 11. **Missing 12:** "...meets <missing 12> again at \( F' \neq D \)." This circle meets \( \Omega \) (the circumcircle), so <missing 12> is **Expression 11: \( \Omega \).** 12. **Missing 13:** "Let <missing 13> meets..." The line \( BF' \) is being discussed, so <missing 13> is **Expression 12: \( BF' \).** 13. **Missing 14:** "...meets <missing 14> again at \( E' \neq F' \)." The line meets \( \Omega \), so <missing 14> is **Expression 11: \( \Omega \).** 14. **Missing 15:** Reim's theorem is used for circles and lines mentioned earlier, which involves \( \omega \) and \( \Omega \), corresponding to **Expression 1: \( \omega, \Omega \).** 15. **Missing 16:** The result derived is about triangle \( ABF \), so <missing 16> is **Expression 16: \( \triangle ABF \).** 16. **Missing 17:** The solution continues with "<missing 17> The point...", referring back to \( \omega \), so <missing 17> is **Expression 17: \( \omega \).** 17. **Missing 18 and 19:** "The point <missing 18> is symmetric to \( D \) with respect to <missing 19>." Here, \( E' \) is symmetric to \( D \) with respect to \( AH \), which are **Expression 10: \( E' \)** and **Expression 9: \( AH \perp BC \implies AH \perp E'D \implies \), respectively. **Answer:** 15, 5, 8, 16, 19, 6, 10, 9, 7, 17, 17, 11, 12, 1, 16, 17, 17, 10, 9 |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| kimi | To solve this problem, we need to match the missing tags in the solution with the provided expressions. Let's go through the solution step by step and identify the appropriate expressions for each missing tag. 1. The first missing tag is <missing 1>. This tag appears in the context of the crosspoint of $BS$ and another line through $H$. The expression that fits this context is <expression 7>, which is $\triangle ABC, BHD$. This expression represents the triangle formed by points $A$, $B$, $C$, and the point $H$ on line $BHD$. 2. The second missing tag is <missing 2>. This tag is related to proving a certain condition. The expression that fits this context is <expression 5>, which is $PH = AH$. This expression is used to establish a relationship between points $P$ and $H$. 3. The third missing tag is <missing 3>. This tag is related to the angles formed by arcs on the circle. The expression that fits this context is <expression 3>, which is $\angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF$. This expression helps to establish the relationship between the angles formed by the arcs and the points on the circle. 4. The fourth missing tag is <missing 4>. This tag is related to the incenter of a certain circle. The expression that fits this context is <expression 4>, which is $\omega$. This expression represents the circle $\omega$, which is the circumcircle of triangle $BDL$. 5. The fifth missing tag is <missing 5>. This tag is related to the orthocenter of a certain triangle. The expression that fits this context is <expression 19>, which is $T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H$. This expression helps to establish the orthocenter of the triangle $TSS'$. 6. The sixth missing tag is <missing 6>. This tag is related to the collinearity of points $B$, $G$, and $F$. The expression that fits this context is <expression 6>, which is $AH \implies E' = E \implies F' = F \implies HF = HD$. This expression helps to establish the collinearity of the points. 7. The seventh missing tag is <missing 7>. This tag is related to the symmetry of a certain line with respect to another line. The expression that fits this context is <expression 8>, which is $=\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H$. This expression helps to establish the symmetry of the line. 8. The eighth missing tag is <missing 8>. This tag is related to the symmetry of a certain line with respect to another line. The expression that fits this context is <expression 15>, which is $AS'$. This expression helps to establish the symmetry of the line. 9. The ninth missing tag is <missing 9>. This tag is related to the orthocenter of a certain triangle. The expression that fits this context is <expression 9>, which is $AH \perp BC \implies AH \perp E'D \implies$. This expression helps to establish the orthocenter of the triangle. 10. The tenth missing tag is <missing 10>. This tag is related to a point on a certain line. The expression that fits this context is <expression 10>, which is $E'$. This expression represents the point $E'$ on the line. 11. The eleventh missing tag is <missing 11>. This tag is related to a circle. The expression that fits this context is <expression 11>, which is $\Omega$. This expression represents the circle $\Omega$. 12. The twelfth missing tag is <missing 12>. This tag is related to a line segment. The expression that fits this context is <expression 12>, which is $BF'$. This expression represents the line segment $BF'$. 13. The thirteenth missing tag is <missing 13>. This tag is related to a point on a certain line. The expression that fits this context is <expression 13>, which is $\implies GF$. This expression represents the point $GF$ on the line. 14. The fourteenth missing tag is <missing 14>. This tag is related to a point on a certain line. The |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| minimax | To solve this problem, we need to match the missing parts of the solution with the corresponding expressions provided. Let's go through the solution step by step and match the expressions to the missing tags. 1. The first missing tag is after "Denote the crosspoint of $BS$ and", which should be a line or a point where $BS$ intersects another line or circle. The only expression that fits this context is <expression 5> PH = AH, which indicates that point H is on the line BS and that PH equals AH. 2. The second missing tag is in the sentence "To finishing the solution we need only to prove that", which suggests that we need to prove a certain property or relationship. The most fitting expression here is <expression 6> AH \implies E' = E \implies F' = F \implies HF = HD, which is a statement about the equality of lengths and the symmetry of points, which is a necessary part of the proof. 3. The third missing tag is in the sentence "Denote $F = SS' \cap AC \implies \angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\f |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| wenxinyiyan | <Detailed reasoning> 1. <missing 1> should be filled with the expression related to the crosspoint of $BS$ and a certain line. Given the context, the line is $AH$, the altitude from $A$ to $BC$. Therefore, <missing 1> corresponds to "AH". This is expression 10 since $AH$ is the orthocenter of $\triangle TSS'$ as mentioned later in the solution. 2. <missing 2> is to prove that a certain point lies on $\omega$. The point is $P$, the intersection of $\omega$ and $\Omega$ other than $B$. So, <missing 2> corresponds to "\omega". This is expression 2. 3. <missing 3> requires filling an expression for an angle. Given the context of equal arcs and angles, <missing 3> should be "\angle EBS' = \varphi". This corresponds to expression 3. 4. <missing 4> refers to the triangle whose incenter is being discussed. The incenter is of $\triangle ABS$ since $S'$ is the internal angle bisector of $\angle BAC$. Thus, <missing 4> is "\triangle ABS". However, this exact expression is not in the list. We use the fact that $S'$ is the midpoint of the arc diametrically opposite to $S$ in $\Omega$ which implies it's the incenter of the triangle formed by $A, B$, and the point where the bisector intersects $BC$ (not explicitly given in the formulae but can be inferred). Since no direct match is found, we can leave it as <missing 4>. 5. <missing 5> is about a point being the orthocenter of a triangle. The point is $H$ and the triangle is $TSS'$. So, <missing 5> corresponds to "PH = AH". This suggests that $P, H$, and the intersection of $AS'$ with $BC$ are collinear and $H$ is the orthocenter of $\triangle TSS'$. The closest expression is 5 since it mentions $PH = AH$. 6. <missing 6> is a logical deduction about points being collinear. Given the lemma and the proof, <missing 6> should be "AH \implies E' = E \implies F' = F \implies HF = HD". This is expression 6. 7. <missing 7> discusses collinearity of points related to a triangle. The points are $B, G$, and $F$ and the triangle is $\triangle ABC$. So, <missing 7> corresponds to "\triangle ABC, BHD". However, the exact collinearity is not directly given in the formulae. The closest related to collinearity is expression 7 which mentions $\triangle ABC$ and $BHD$ but not directly their collinearity. We can infer it from the proof though. 8. <missing 8> seems to be about symmetry with respect to a line. Given the context of the lemma, <missing 8> should be "=\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H". This is a part of expression 3. 9-19. The remaining <missing> tags are either not directly related to the given formulae or are too specific to be filled without the exact context of the complete solution. Since they are not critical to the main reasoning, we can leave them as is. Answer: 10, 2, 3, 5, 6, 7 (with some inferences and assumptions for 4 and 7) |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| zhipu | ### Detailed Reasoning 1. **Identifying <missing 1>**: - The context suggests that <missing 1> should be a line or circle that intersects with $BS$. Given the context of the problem, it is likely referring to the line $AS'$, which is the internal angle bisector of $\angle BAC$. - **Best Match**: <expression 15> AS' 2. **Identifying <missing 2>**: - The solution indicates that we need to prove something about $AH$. The next step involves proving a relationship involving $AH$ and other points. - **Best Match**: <expression 5> PH = AH. 3. **Identifying <missing 3>**: - This part of the solution is discussing angles and their relationships. Specifically, it mentions $\angle CBS$ and its relation to other angles. - **Best Match**: <expression 8> =\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H 4. **Identifying <missing 4>**: - The context suggests that <missing 4> should be a triangle or a figure related to the incenter. - **Best Match**: <expression 16> \triangle ABF. 5. **Identifying <missing 5>**: - This part is discussing the orthocenter of a triangle. - **Best Match**: <expression 19> T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H 6. **Identifying <missing 6>**: - This part is discussing points being collinear. - **Best Match**: <expression 13> \implies GF 7. **Identifying <missing 7>**: - This part is discussing symmetry with respect to a line. - **Best Match**: <expression 7> \triangle ABC, BHD 8. **Identifying <missing 8>**: - This part is discussing symmetry with respect to a point or line. - **Best Match**: <expression 18> E'D || BC 9. **Identifying <missing 9>**: - This part is discussing the orthocenter of a triangle. - **Best Match**: <expression 17> \omega 10. **Identifying <missing 10>**: - This part is discussing a circle centered at $H$. - **Best Match**: <expression 11> \Omega 11. **Identifying <missing 11>**: - This part is discussing a circle meeting another circle. - **Best Match**: <expression 12> BF' 12. **Identifying <missing 12>**: - This part is discussing a circle meeting another circle. - **Best Match**: <expression 14> TF. 13. **Identifying <missing 13>**: - This part is discussing a line meeting another line. - **Best Match**: <expression 10> E' 14. **Identifying <missing 14>**: - This part is discussing a line meeting another line. - **Best Match**: <expression 9> AH \perp BC \implies AH \perp E'D \implies 15. **Identifying <missing 15>**: - This part is discussing a theorem involving lines. - **Best Match**: <expression 3> \angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies 16. **Identifying <missing 16>**: - This part is discussing a theorem involving triangles. - **Best Match**: <expression 6> AH \implies E' = E \implies F' = F \implies HF = HD. 17. **Identifying <missing 17>**: - This part is discussing a point being symmetric. - **Best Match**: <expression 4> \omega 18. **Identifying <missing 18>**: - This part is discussing a point being symmetric. - **Best Match**: <expression 1> \omega, \Omega 19. **Identifying <missing 19>**: - This part is discussing symmetry with respect to a line. - **Best Match**: <expression 2> \omega ### Final Answer Answer: 15, 5, 8, 16, 19, 13, 7, 18, 17, 11, 12, 14, 10, 9, 3, 6, 4, 1, 2 |
15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |
| deepseek-v3 | Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-d4c0e591-5396-9e46-8407-def493b79e0e', 'request_id': 'd4c0e591-5396-9e46-8407-def493b79e0e'} | 15,5,8,16,19,3,13,14,7,4,17,11,2,12,1,18,9,10,6 |