金声玉亮2.0模型评测报告

问题

You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format "<expression Y> = $<latex code>$" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>.

The question is:
Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.

The solution is:
Denote the point diametrically opposite to a point $S$ through $S' \implies AS'$ is the internal angle bisector of $\angle BAC$ .

Denote the crosspoint of $BS$ and $AS'$ through $H, \angle ABS = \varphi.$ \[AE \perp BC, SS' \perp BC \implies \overset{\Large\frown} {AS} = \overset{\Large\frown} {ES'} = 2\varphi \implies\]
\[\angle EAS' = \varphi = \angle ABS \implies \angle DAH = \angle ABH \implies\]\[\triangle AHD \sim \triangle BAH \implies \frac {AH}{BH} = \frac {DH}{AH} \implies AH^2 = BH \cdot DH.\]To finishing the solution we need only to prove that $PH = AH.$ Denote $F = SS' \cap AC \implies \angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\frown} {BS}}{2} = \frac {\overset{\Large\frown} {AB}}{2} + \frac {\overset{\Large\frown} {AS}}{2} =$ $=\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H$ is incenter of $\triangle ABF.$ Denote $T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H$ is the orthocenter of $\triangle TSS'.$ Denote $G = PS' \cap AE \implies \angle BPG = \angle BPS' = \angle BSS' = \angle BDG \implies B, L, P, D,$ and $G$ are concyclic.

2023 IMO 2 lemma.png $\angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies$ points $B, G,$ and $F$ are collinear $\implies GF$ is symmetric to $AF$ with respect $TF.$ We use the lemma and complete the proof.

Lemma 1

2023 IMO 2b Lemma.png
Let acute triangle $\triangle ABC, AB > AC$ be given.

Let $H$ be the orthocenter of $\triangle ABC, BHD$ be the height.

Let $\Omega$ be the circle $BCD. BC$ is the diameter of <missing 1> The point $E$ is symmetric to $D$ with respect to $AH.$ The line $BE$ meets $\Omega$ again at $F \neq B$ .

Prove that $HF = HD.$ Proof

Let $\omega$ be the circle centered at $H$ with radius $HD.$ The $\omega$ meets $\Omega$ again at $F' \neq D, HD = HF'.$ Let $\omega$ meets $BF'$ again at $E' \neq F'$ .

We use Reim’s theorem for $\omega, \Omega$ and lines $CDD$ and <missing 2> and get $E'D || BC$ (this idea was recommended by Leonid Shatunov). $AH \perp BC \implies AH \perp E'D \implies$ The point $E'$ is symmetric to $D$ with respect to <missing 3>

The formulae are:
<expression 1> AH \implies E' = E \implies F' = F \implies HF = HD.
<expression 2> \Omega.
<expression 3> BE'F'

Your final answer should be STRICTLY in the format:

<Detailed reasoning>

Answer: <comma separated list of numbers representing expression identifiers>

答案

模型	模型答案	正确答案
金声玉亮2.0	To match the formulae to the missing tags in the solution, let's analyze the solution step by step and identify which formulae are needed to fill the gaps marked by <missing X>. 1. <missing 1>: The text mentions "BC is the diameter of <missing 1>". The context suggests that this refers to the circle \(\Omega\), which is the circumcircle of triangle \(BCD\). Therefore, the missing expression should be \(\Omega\). 2. <missing 2>: The text states "and lines \(CDD\) and <missing 2>". This refers to the lines involved in the Reim’s theorem application. The lines \(CDD\) and the other line should be part of the theorem. Given the context, the other line should be \(BE'F'\) as it is the line that intersects \(\Omega\) again at \(F'\). 3. <missing 3>: The text mentions "The point \(E'\) is symmetric to \(D\) with respect to <missing 3>". This refers to the line or axis of symmetry. Given the context, the axis of symmetry should be \(AH\) as it is the line through the orthocenter \(H\) and perpendicular to \(BC\). Now, let's match these to the expressions provided: - <expression 1>: \(AH \implies E' = E \implies F' = F \implies HF = HD\). This expression is not directly needed to fill any of the missing tags. - <expression 2>: \(\Omega\). This matches <missing 1>. - <expression 3>: \(BE'F'\). This matches <missing 2>. Thus, the correct mapping is: - <missing 1> is filled by <expression 2> - <missing 2> is filled by <expression 3> - <missing 3> is filled by <expression 1> Answer: 2, 3, 1	2,3,1
deepseek-r1	<Detailed reasoning> For Lemma 1, the first missing tag (<missing 1>) refers to the circle of which BC is the diameter. Since Ω is defined as the circle passing through B, C, and D, and BC being the diameter implies Ω, hence expression 2. For Reim's theorem, the lines involved are CD and BE'F' (a line through points B, E', F'), so <missing 2> is expression 3. Finally, the symmetry of E' and D is with respect to AH (as AH is perpendicular to E'D), so <missing 3> is expression 1. </Detailed reasoning> Answer: 2,3,1	2,3,1
gpto1	To solve this problem, we need to match the given expressions to the missing parts in the solution. Let's analyze each missing part and determine which expression fits best. Missing Part 1: Text: Let $\Omega$ be the circle $BCD$. $BC$ is the diameter of <missing 1> We are told that $\Omega$ is the circle passing through points $B$, $C$, and $D$. Since $BC$ is a chord of circle $\Omega$, and it's stated that $BC$ is a diameter, it makes sense that <missing 1> is $\Omega$ itself. Corresponding Expression: - Expression 2: $\Omega$ Missing Part 2: Text: We use Reim’s theorem for $\omega$, $\Omega$ and lines $CDD$ and <missing 2> and get $E'D \|\| BC$. Reim's theorem relates two circles and two lines intersecting them. Here, we have circles $\omega$ and $\Omega$, lines $CDD$, and the missing line. Since $E'$ and $F'$ are points where $\omega$ and $\Omega$ intersect, and $B$ is a common point, the line we're missing is the line through $B$, $E'$, and $F'$. Corresponding Expression: - Expression 1: $BE'F'$ Missing Part 3: Text: The point $E'$ is symmetric to $D$ with respect to <missing 3> Given that $AH \perp E'D$ and $AH \perp BC$, it implies that $AH$ is the line of symmetry between $D$ and $E'$. Therefore, <missing 3> should be $AH$. Corresponding Expression: - Expression 3: $AH \implies E' = E \implies F' = F \implies HF = HD$ Final Answer: Matching the expressions to the missing parts, we get: 1. <missing 1> = Expression 2 2. <missing 2> = Expression 1 3. <missing 3> = Expression 3 Answer: 2,1,3	2,3,1
gpt-4o-2024-08-06	To solve this problem, we need to match the missing expressions in the solution with the given formulae. Let's analyze the solution step by step: 1. Understanding the context of the missing parts: - <missing 1>: The text mentions "BC is the diameter of <missing 1>". This suggests that the missing part should be a circle or a geometric figure where BC is a diameter. In the context of the problem, this is likely referring to the circumcircle of triangle BCD, which is denoted as \(\Omega\). - <missing 2>: The text mentions using Reim’s theorem for \(\omega\), \(\Omega\), and lines \(CDD\) and <missing 2>. Reim's theorem typically involves two circles and two lines, where the lines are parallel. The missing part should be a line that is parallel to \(CDD\). Given the context, this line is likely \(BE'F'\), which is mentioned in the expressions. - <missing 3>: The text states "The point \(E'\) is symmetric to \(D\) with respect to <missing 3>". This suggests that the missing part is a line or axis of symmetry. Since \(E'\) is symmetric to \(D\) with respect to \(AH\), it makes sense that the missing part is \(AH\). 2. Matching the expressions: - Expression 1: \(AH \implies E' = E \implies F' = F \implies HF = HD.\) This expression involves \(AH\), which is the axis of symmetry for \(E'\) and \(D\). Therefore, it matches <missing 3>. - Expression 2: \(\Omega.\) This is the circumcircle of triangle BCD, which matches <missing 1>. - Expression 3: \(BE'F'\) is a line, and in the context of Reim's theorem, it is likely the line parallel to \(CDD\). Therefore, it matches <missing 2>. 3. Conclusion: Based on the analysis, the missing expressions are matched as follows: - <missing 1> is matched with Expression 2 (\(\Omega\)). - <missing 2> is matched with Expression 3 (\(BE'F'\)). - <missing 3> is matched with Expression 1 (\(AH\)). Answer: 2, 3, 1	2,3,1
DeepSeek-Coder-V2-Lite-Instruct	<Detailed reasoning> First, let's identify the missing parts in the solution. The solution uses several geometric properties and theorems, and we need to match these with the given formulae. 1. Missing Tag 1: The circumcircle of triangle \(BCD\) is mentioned, and \(BC\) is the diameter of this circle. This implies that the circle is the circumcircle of triangle \(BCD\). 2. Missing Tag 2: The line \(E'D\) is parallel to \(BC\), and this is derived from a geometric theorem applied to the circles \(\omega\) and \(\Omega\). 3. Missing Tag 3: The point \(E'\) is symmetric to \(D\) with respect to the perpendicular bisector of \(BE'\). Now, let's match these with the given formulae: - Expression 1: This refers to the proof that \(HF = HD\) using Reim’s theorem. - Expression 2: This refers to the circumcircle \(\Omega\) of triangle \(ABC\). - Expression 3: This refers to the line \(BE'F'\) and the symmetry of point \(E'\) with respect to the perpendicular bisector of \(BE'\). Given these matches, the corresponding expression identifiers are: 1. Expression 1 corresponds to <missing 1>. 2. Expression 2 corresponds to <missing 2>. 3. Expression 3 corresponds to <missing 3>. Thus, the list of numbers representing the expression identifiers is: Answer: 1, 2, 3	2,3,1
Meta-Llama-3.1-405B-Instruct-Turbo	To solve this problem, we need to match the formulae to the missing tags in the solution. The first missing tag is <missing 1>. The sentence is "BC is the diameter of <missing 1>." This suggests that the answer should be the identifier of the expression that represents the circle with diameter BC. The formulae are given as "<expression Y> = $<latex code>$", but the latex code is not provided for the expressions. However, based on the context, we can infer that <expression 2> = $\Omega$ is the circle with diameter BC. Therefore, the answer for <missing 1> is 2. The second missing tag is <missing 2>. The sentence is "We use Reim’s theorem for $\omega, \Omega$ and lines $CDD$ and <missing 2>...". This suggests that the answer should be the identifier of the expression that represents a line. Based on the context, we can infer that <expression 3> = $BE'F'$ is a line. Therefore, the answer for <missing 2> is 3. The third missing tag is <missing 3>. The sentence is "...The point $E'$ is symmetric to $D$ with respect to <missing 3>." This suggests that the answer should be the identifier of the expression that represents a line or a point of symmetry. Based on the context, we can infer that <expression 1> = $AH$ is a line of symmetry. Therefore, the answer for <missing 3> is 1. Therefore, the final answer is: Answer: 2, 3, 1	2,3,1
Meta-Llama-3.1-70B-Instruct-Turbo	To solve this problem, we need to match the formulae to the missing tags in the solution. Let's analyze the solution step by step. The first missing tag is <missing 1>, which is related to the circle $\Omega$. The solution states that "BC is the diameter of <missing 1>". Looking at the formulae, we can see that <expression 2> is $\Omega$, which is the circle. Therefore, <missing 1> is likely to be $\Omega$. The second missing tag is <missing 2>, which is related to the lines $CDD$ and another line. The solution states that "We use Reim’s theorem for $\omega, \Omega$ and lines $CDD$ and <missing 2>". Looking at the formulae, we can see that <expression 3> is $BE'F'$, which is a line. Therefore, <missing 2> is likely to be $BE'F'$. The third missing tag is <missing 3>, which is related to the point $E'$ being symmetric to $D$ with respect to a certain line. The solution states that "The point $E'$ is symmetric to $D$ with respect to <missing 3>". Looking at the formulae, we can see that <expression 1> is $AH$, which is a line. Therefore, <missing 3> is likely to be $AH$. Based on this analysis, our answer is: Answer: 2, 3, 1	2,3,1
Qwen2-72B-Instruct	<Detailed reasoning> In the solution, the first missing part refers to the circle that has BC as its diameter. This is mentioned right after "Let $\Omega$ be the circle $BCD. BC$ is the diameter of <missing 1>". Therefore, <missing 1> corresponds to the description of the circle with BC as its diameter, which is expression 2. The second missing part is within the context of applying Reim's theorem. It states, "We use Reim’s theorem for $\omega, \Omega$ and lines $CDD$ and <missing 2>". Since Reim's theorem involves circles and lines, and we already have $\omega$ and $\Omega$, the missing part must refer to a line. Given the context, it seems like the line being referred to is BE', which is described in expression 3. The third missing part is within the sentence, "AH \perp E'D \implies The point $E'$ is symmetric to $D$ with respect to <missing 3>". Here, the symmetry of E' with respect to D is being discussed, and since AH is perpendicular to E'D, it implies that AH is the line of symmetry. This matches the description in expression 1. Answer: 2, 3, 1	2,3,1
baichuan4	Detailed reasoning: 1. The first missing tag, <missing 1>, refers to the circle mentioned in the lemma. Since the lemma discusses the circle $\Omega$ with $BC$ as its diameter, it is clear that <missing 1> should be filled with the reference to this circle. Therefore, <missing 1> corresponds to <expression 2>. 2. The second missing tag, <missing 2>, appears in the context of applying Reim's theorem. Reim's theorem typically involves parallel lines and circles, so the missing line here would be one that is parallel to another significant line in the configuration. Given that $E'D$ is already mentioned to be parallel to $BC$, it seems likely that <missing 2> refers to the line $BE'F'$. Hence, <missing 2> corresponds to <expression 3>. 3. The third missing tag, <missing 3>, is in the context where $E'$ is described as being symmetric to $D$ with respect to a certain line. Given the configuration, it is most likely that this line is $AH$, which is perpendicular to $BC$ and $E'D$. Thus, <missing 3> corresponds to <expression 1>. Answer: 2, 3, 1	2,3,1
doubao	<Detailed reasoning>: In the solution, for the first missing part, the circle that has BC as the diameter is described, which is related to expression 1. For the second missing part, the other circle involved in Reim's theorem is the original circle Ω, which is related to expression 2. For the third missing part, the symmetry is with respect to the line BE', which is related to expression 3. Answer: 1, 2, 3	2,3,1
gpto1	To solve this problem, we need to match the given expressions to the missing parts in the solution. Let's analyze each missing part and determine which expression fits best. Missing Part 1: Text: Let $\Omega$ be the circle $BCD$. $BC$ is the diameter of <missing 1> We are told that $\Omega$ is the circle passing through points $B$, $C$, and $D$. Since $BC$ is a chord of circle $\Omega$, and it's stated that $BC$ is a diameter, it makes sense that <missing 1> is $\Omega$ itself. Corresponding Expression: - Expression 2: $\Omega$ Missing Part 2: Text: We use Reim’s theorem for $\omega$, $\Omega$ and lines $CDD$ and <missing 2> and get $E'D \|\| BC$. Reim's theorem relates two circles and two lines intersecting them. Here, we have circles $\omega$ and $\Omega$, lines $CDD$, and the missing line. Since $E'$ and $F'$ are points where $\omega$ and $\Omega$ intersect, and $B$ is a common point, the line we're missing is the line through $B$, $E'$, and $F'$. Corresponding Expression: - Expression 1: $BE'F'$ Missing Part 3: Text: The point $E'$ is symmetric to $D$ with respect to <missing 3> Given that $AH \perp E'D$ and $AH \perp BC$, it implies that $AH$ is the line of symmetry between $D$ and $E'$. Therefore, <missing 3> should be $AH$. Corresponding Expression: - Expression 3: $AH \implies E' = E \implies F' = F \implies HF = HD$ Final Answer: Matching the expressions to the missing parts, we get: 1. <missing 1> = Expression 2 2. <missing 2> = Expression 1 3. <missing 3> = Expression 3 Answer: 2,1,3	2,3,1
kimi	<Detailed reasoning> To match the formulae to the missing tags in the solution, I will analyze the context and content of each missing tag and compare it to the given formulae. <missing 1>: The context suggests that we are talking about a circle with diameter BC. The formula that matches this context is <expression 2>, which states that BC is the diameter of the circle Ω. <missing 2>: The context mentions using Reim's theorem for the circles and lines CDD and another line. The formula that matches this context is <expression 3>, which states that the line BE'F' is involved in the application of Reim's theorem. <missing 3>: The context talks about the point E' being symmetric to D with respect to a certain line. The formula that matches this context is <expression 1>, which states that AH is the line with respect to which E' is symmetric to D. Answer: 2, 3, 1	2,3,1
minimax	To solve this problem, we need to match the missing parts of the solution with the corresponding formulae. Let's go through the solution step by step and see where each formula fits. 1. The first missing part is in the sentence "BC is the diameter of <missing 1>." This is clearly referring to the circle $\Omega$, which is the circumcircle of triangle $ABC$. Therefore, the formula that fits here is the one that describes the circumcircle. 2. The second missing part is in the sentence "Reim’s theorem for $\omega, \Omega$ and lines $CDD$ and <missing 2>." Since Reim's theorem is used to show that $E'D \|\| BC$, and we know that $AH \perp BC$, it is logical that the missing part refers to the line $AH$. This is because the theorem is applied to the circles $\omega$ and $\Omega$ and the line $AH$. 3. The third missing part is in the sentence "$AH \perp BC \implies AH \perp E'D \implies$ The point $E'$ is symmetric to $D$ with respect to <missing 3>." Since $E'$ is symmetric to $D$ with respect	2,3,1
wenxinyiyan	Detailed reasoning: First, let's analyze the solution step by step to understand what is missing. 1. The solution talks about a point diametrically opposite to $S$ through $S'$, and it is mentioned that $AS'$ is the internal angle bisector of $\angle BAC$. 2. Then, it is mentioned that $BS$ and $AS'$ meet at $H$, and $\angle ABS = \varphi$. 3. After some angle calculations, it is shown that $\triangle AHD \sim \triangle BAH$. 4. The solution then claims that to finish the proof, we need to show that $PH = AH$. 5. Some more constructions and angle calculations follow, leading to the conclusion that $B, L, P, D,$ and $G$ are concyclic. 6. Finally, the solution talks about a lemma and uses it to complete the proof. Now, let's look at the missing parts and try to match them with the given formulae. <missing 1>: This should be related to the circle $\Omega$ and its properties. Looking at the formulae, <expression 2> talks about $\Omega$ being the circle $BCD$ with $BC$ as the diameter. This seems to fit here as it describes the properties of $\Omega$ that are used in the solution. <missing 2>: This part is about applying Reim's theorem to $\omega, \Omega$ and lines $CDD$ and another line. Among the formulae, <expression 3> talks about lines $BE'F'$ which could be the other line mentioned in the missing part. <missing 3>: This should be related to the symmetry of points with respect to a line. Among the formulae, <expression 1> talks about $AH$ being the line of symmetry for points $E'$ and $D$. This seems to fit here as it explains the symmetry property used in the solution. Answer: 2, 3, 1	2,3,1
zhipu	### Detailed Reasoning 1. Understanding the Context of the Missing Tags: - <missing 1>: This is in the context of describing the circle $\Omega$ where $BC$ is the diameter. We need a formula that fits this description. - <missing 2>: This is used in the application of Reim’s theorem, involving the circles $\omega$ and $\Omega$ and the lines $CD$ and another line. We need a formula that describes a relevant line or property. - <missing 3>: This is in the context of symmetry with respect to a line or point. We need a formula that fits this description. 2. Analyzing the Given Formulae: - <expression 1>: `AH \implies E' = E \implies F' = F \implies HF = HD.` This seems to be a conclusion or a sequence of implications rather than a descriptive formula for a circle or line. - <expression 2>: `\Omega.` This directly refers to the circle $\Omega$. - <expression 3>: `BE'F'`. This refers to a set of points or a line involving $B$, $E'$, and $F'$. 3. Matching the Formulae to the Missing Tags: - <missing 1>: The description "BC is the diameter of <missing 1>" suggests we need something that describes the circle $\Omega$. The most fitting formula here is `<expression 2> \Omega.` because it directly refers to the circle $\Omega$. - <missing 2>: In the context of Reim’s theorem, we need a line or a set of points that are relevant. `<expression 3> BE'F'` seems to fit this context as it involves points on the circle and lines related to $B$, $E'$, and $F'$. - <missing 3>: The context is about symmetry with respect to a line or point. `<expression 1> AH \implies E' = E \implies F' = F \implies HF = HD.` seems to describe a symmetry or equivalence, fitting the context of symmetry. ### Final Answer Answer: 2, 3, 1	2,3,1
deepseek-v3	<Detailed reasoning> 1. Identifying <missing 1>: In the lemma, the sentence "BC is the diameter of <missing 1>" suggests that <missing 1> should be the circle for which BC is the diameter. Given the context, the circle in question is \(\Omega\), which is the circle \(BCD\). Therefore, <missing 1> should be replaced by \(\Omega\). The corresponding expression is <expression 2>. 2. Identifying <missing 2>: In the lemma, the sentence "We use Reim’s theorem for \(\omega\), \(\Omega\) and lines \(CDD\) and <missing 2>" suggests that <missing 2> should be the line that is being compared to \(CDD\). Given the context, this line is \(BE'F'\). Therefore, <missing 2> should be replaced by \(BE'F'\). The corresponding expression is <expression 3>. 3. Identifying <missing 3>: In the lemma, the sentence "The point \(E'\) is symmetric to \(D\) with respect to <missing 3>" suggests that <missing 3> should be the line or point with respect to which \(E'\) is symmetric to \(D\). Given the context, this is \(AH\). Therefore, <missing 3> should be replaced by \(AH\). The corresponding expression is <expression 1>. Answer: 2, 3, 1	2,3,1

模型

模型答案

正确答案

金声玉亮2.0

To match the formulae to the missing tags in the solution, let's analyze the solution step by step and identify which formulae are needed to fill the gaps marked by <missing X>.

1. **<missing 1>**: The text mentions "BC is the diameter of <missing 1>". The context suggests that this refers to the circle $\Omega$, which is the circumcircle of triangle $BCD$. Therefore, the missing expression should be $\Omega$.

2. **<missing 2>**: The text states "and lines $CDD$ and <missing 2>". This refers to the lines involved in the Reim’s theorem application. The lines $CDD$ and the other line should be part of the theorem. Given the context, the other line should be $BE'F'$ as it is the line that intersects $\Omega$ again at $F'$.

3. **<missing 3>**: The text mentions "The point $E'$ is symmetric to $D$ with respect to <missing 3>". This refers to the line or axis of symmetry. Given the context, the axis of symmetry should be $AH$ as it is the line through the orthocenter $H$ and perpendicular to $BC$.

Now, let's match these to the expressions provided:

- **<expression 1>**: $AH \implies E' = E \implies F' = F \implies HF = HD$. This expression is not directly needed to fill any of the missing tags.
- **<expression 2>**: $\Omega$. This matches <missing 1>.
- **<expression 3>**: $BE'F'$. This matches <missing 2>.

Thus, the correct mapping is:
- <missing 1> is filled by <expression 2>
- <missing 2> is filled by <expression 3>
- <missing 3> is filled by <expression 1>

Answer: 2, 3, 1

2,3,1

deepseek-r1

<Detailed reasoning>
For Lemma 1, the first missing tag (<missing 1>) refers to the circle of which BC is the diameter. Since Ω is defined as the circle passing through B, C, and D, and BC being the diameter implies Ω, hence expression 2. For Reim's theorem, the lines involved are CD and BE'F' (a line through points B, E', F'), so <missing 2> is expression 3. Finally, the symmetry of E' and D is with respect to AH (as AH is perpendicular to E'D), so <missing 3> is expression 1.
</Detailed reasoning>

Answer: 2,3,1

2,3,1

gpto1

To solve this problem, we need to match the given expressions to the missing parts in the solution. Let's analyze each missing part and determine which expression fits best.

**Missing Part 1:**
*Text:* Let $\Omega$ be the circle $BCD$. $BC$ is the diameter of **<missing 1>**

We are told that $\Omega$ is the circle passing through points $B$, $C$, and $D$. Since $BC$ is a chord of circle $\Omega$, and it's stated that $BC$ is a diameter, it makes sense that **<missing 1>** is $\Omega$ itself.

*Corresponding Expression:*
- **Expression 2:** $\Omega$

**Missing Part 2:**
*Text:* We use Reim’s theorem for $\omega$, $\Omega$ and lines $CDD$ and **<missing 2>** and get $E'D || BC$.

Reim's theorem relates two circles and two lines intersecting them. Here, we have circles $\omega$ and $\Omega$, lines $CDD$, and the missing line. Since $E'$ and $F'$ are points where $\omega$ and $\Omega$ intersect, and $B$ is a common point, the line we're missing is the line through $B$, $E'$, and $F'$.

*Corresponding Expression:*
- **Expression 1:** $BE'F'$

**Missing Part 3:**
*Text:* The point $E'$ is symmetric to $D$ with respect to **<missing 3>**

Given that $AH \perp E'D$ and $AH \perp BC$, it implies that $AH$ is the line of symmetry between $D$ and $E'$. Therefore, **<missing 3>** should be $AH$.

*Corresponding Expression:*
- **Expression 3:** $AH \implies E' = E \implies F' = F \implies HF = HD$

**Final Answer:** Matching the expressions to the missing parts, we get:

1. **<missing 1>** = Expression **2**
2. **<missing 2>** = Expression **1**
3. **<missing 3>** = Expression **3**

**Answer:** 2,1,3

2,3,1

gpt-4o-2024-08-06

To solve this problem, we need to match the missing expressions in the solution with the given formulae. Let's analyze the solution step by step:

1. **Understanding the context of the missing parts:**

- **<missing 1>:** The text mentions "BC is the diameter of <missing 1>". This suggests that the missing part should be a circle or a geometric figure where BC is a diameter. In the context of the problem, this is likely referring to the circumcircle of triangle BCD, which is denoted as $\Omega$.

- **<missing 2>:** The text mentions using Reim’s theorem for $\omega$, $\Omega$, and lines $CDD$ and <missing 2>. Reim's theorem typically involves two circles and two lines, where the lines are parallel. The missing part should be a line that is parallel to $CDD$. Given the context, this line is likely $BE'F'$, which is mentioned in the expressions.

- **<missing 3>:** The text states "The point $E'$ is symmetric to $D$ with respect to <missing 3>". This suggests that the missing part is a line or axis of symmetry. Since $E'$ is symmetric to $D$ with respect to $AH$, it makes sense that the missing part is $AH$.

2. **Matching the expressions:**

- **Expression 1:** $AH \implies E' = E \implies F' = F \implies HF = HD.$ This expression involves $AH$, which is the axis of symmetry for $E'$ and $D$. Therefore, it matches <missing 3>.

- **Expression 2:** $\Omega.$ This is the circumcircle of triangle BCD, which matches <missing 1>.

- **Expression 3:** $BE'F'$ is a line, and in the context of Reim's theorem, it is likely the line parallel to $CDD$. Therefore, it matches <missing 2>.

3. **Conclusion:**

Based on the analysis, the missing expressions are matched as follows:
- <missing 1> is matched with Expression 2 ($\Omega$).
- <missing 2> is matched with Expression 3 ($BE'F'$).
- <missing 3> is matched with Expression 1 ($AH$).

Answer: 2, 3, 1

2,3,1

DeepSeek-Coder-V2-Lite-Instruct

<Detailed reasoning>

First, let's identify the missing parts in the solution. The solution uses several geometric properties and theorems, and we need to match these with the given formulae.

1. **Missing Tag 1**: The circumcircle of triangle $BCD$ is mentioned, and $BC$ is the diameter of this circle. This implies that the circle is the circumcircle of triangle $BCD$.
2. **Missing Tag 2**: The line $E'D$ is parallel to $BC$, and this is derived from a geometric theorem applied to the circles $\omega$ and $\Omega$.
3. **Missing Tag 3**: The point $E'$ is symmetric to $D$ with respect to the perpendicular bisector of $BE'$.

Now, let's match these with the given formulae:

- **Expression 1**: This refers to the proof that $HF = HD$ using Reim’s theorem.
- **Expression 2**: This refers to the circumcircle $\Omega$ of triangle $ABC$.
- **Expression 3**: This refers to the line $BE'F'$ and the symmetry of point $E'$ with respect to the perpendicular bisector of $BE'$.

Given these matches, the corresponding expression identifiers are:

1. Expression 1 corresponds to <missing 1>.
2. Expression 2 corresponds to <missing 2>.
3. Expression 3 corresponds to <missing 3>.

Thus, the list of numbers representing the expression identifiers is:

Answer: 1, 2, 3

2,3,1

Meta-Llama-3.1-405B-Instruct-Turbo

To solve this problem, we need to match the formulae to the missing tags in the solution.

The first missing tag is <missing 1>. The sentence is "BC is the diameter of <missing 1>." This suggests that the answer should be the identifier of the expression that represents the circle with diameter BC.

The formulae are given as "<expression Y> = $<latex code>$", but the latex code is not provided for the expressions. However, based on the context, we can infer that <expression 2> = $\Omega$ is the circle with diameter BC. Therefore, the answer for <missing 1> is 2.

The second missing tag is <missing 2>. The sentence is "We use Reim’s theorem for $\omega, \Omega$ and lines $CDD$ and <missing 2>...". This suggests that the answer should be the identifier of the expression that represents a line.

Based on the context, we can infer that <expression 3> = $BE'F'$ is a line. Therefore, the answer for <missing 2> is 3.

The third missing tag is <missing 3>. The sentence is "...The point $E'$ is symmetric to $D$ with respect to <missing 3>." This suggests that the answer should be the identifier of the expression that represents a line or a point of symmetry.

Based on the context, we can infer that <expression 1> = $AH$ is a line of symmetry. Therefore, the answer for <missing 3> is 1.

Therefore, the final answer is:

Answer: 2, 3, 1

2,3,1

Meta-Llama-3.1-70B-Instruct-Turbo

To solve this problem, we need to match the formulae to the missing tags in the solution. Let's analyze the solution step by step.

The first missing tag is <missing 1>, which is related to the circle $\Omega$. The solution states that "BC is the diameter of <missing 1>". Looking at the formulae, we can see that <expression 2> is $\Omega$, which is the circle. Therefore, <missing 1> is likely to be $\Omega$.

The second missing tag is <missing 2>, which is related to the lines $CDD$ and another line. The solution states that "We use Reim’s theorem for $\omega, \Omega$ and lines $CDD$ and <missing 2>". Looking at the formulae, we can see that <expression 3> is $BE'F'$, which is a line. Therefore, <missing 2> is likely to be $BE'F'$.

The third missing tag is <missing 3>, which is related to the point $E'$ being symmetric to $D$ with respect to a certain line. The solution states that "The point $E'$ is symmetric to $D$ with respect to <missing 3>". Looking at the formulae, we can see that <expression 1> is $AH$, which is a line. Therefore, <missing 3> is likely to be $AH$.

Based on this analysis, our answer is:

Answer: 2, 3, 1

2,3,1

Qwen2-72B-Instruct

<Detailed reasoning>

In the solution, the first missing part refers to the circle that has BC as its diameter. This is mentioned right after "Let $\Omega$ be the circle $BCD. BC$ is the diameter of <missing 1>". Therefore, <missing 1> corresponds to the description of the circle with BC as its diameter, which is expression 2.

The second missing part is within the context of applying Reim's theorem. It states, "We use Reim’s theorem for $\omega, \Omega$ and lines $CDD$ and <missing 2>". Since Reim's theorem involves circles and lines, and we already have $\omega$ and $\Omega$, the missing part must refer to a line. Given the context, it seems like the line being referred to is BE', which is described in expression 3.

The third missing part is within the sentence, "AH \perp E'D \implies The point $E'$ is symmetric to $D$ with respect to <missing 3>". Here, the symmetry of E' with respect to D is being discussed, and since AH is perpendicular to E'D, it implies that AH is the line of symmetry. This matches the description in expression 1.

Answer: 2, 3, 1

2,3,1

baichuan4

Detailed reasoning:

1. The first missing tag, <missing 1>, refers to the circle mentioned in the lemma. Since the lemma discusses the circle $\Omega$ with $BC$ as its diameter, it is clear that <missing 1> should be filled with the reference to this circle. Therefore, <missing 1> corresponds to <expression 2>.

2. The second missing tag, <missing 2>, appears in the context of applying Reim's theorem. Reim's theorem typically involves parallel lines and circles, so the missing line here would be one that is parallel to another significant line in the configuration. Given that $E'D$ is already mentioned to be parallel to $BC$, it seems likely that <missing 2> refers to the line $BE'F'$. Hence, <missing 2> corresponds to <expression 3>.

3. The third missing tag, <missing 3>, is in the context where $E'$ is described as being symmetric to $D$ with respect to a certain line. Given the configuration, it is most likely that this line is $AH$, which is perpendicular to $BC$ and $E'D$. Thus, <missing 3> corresponds to <expression 1>.

Answer: 2, 3, 1

2,3,1

doubao

<Detailed reasoning>: In the solution, for the first missing part, the circle that has BC as the diameter is described, which is related to expression 1. For the second missing part, the other circle involved in Reim's theorem is the original circle Ω, which is related to expression 2. For the third missing part, the symmetry is with respect to the line BE', which is related to expression 3.
Answer: 1, 2, 3

2,3,1

gpto1

2,3,1

kimi

<Detailed reasoning>

To match the formulae to the missing tags in the solution, I will analyze the context and content of each missing tag and compare it to the given formulae.

<missing 1>: The context suggests that we are talking about a circle with diameter BC. The formula that matches this context is <expression 2>, which states that BC is the diameter of the circle Ω.

<missing 2>: The context mentions using Reim's theorem for the circles and lines CDD and another line. The formula that matches this context is <expression 3>, which states that the line BE'F' is involved in the application of Reim's theorem.

<missing 3>: The context talks about the point E' being symmetric to D with respect to a certain line. The formula that matches this context is <expression 1>, which states that AH is the line with respect to which E' is symmetric to D.

Answer: 2, 3, 1

2,3,1

minimax

To solve this problem, we need to match the missing parts of the solution with the corresponding formulae. Let's go through the solution step by step and see where each formula fits.

1. The first missing part is in the sentence "BC is the diameter of <missing 1>." This is clearly referring to the circle $\Omega$, which is the circumcircle of triangle $ABC$. Therefore, the formula that fits here is the one that describes the circumcircle.

2. The second missing part is in the sentence "Reim’s theorem for $\omega, \Omega$ and lines $CDD$ and <missing 2>." Since Reim's theorem is used to show that $E'D || BC$, and we know that $AH \perp BC$, it is logical that the missing part refers to the line $AH$. This is because the theorem is applied to the circles $\omega$ and $\Omega$ and the line $AH$.

3. The third missing part is in the sentence "$AH \perp BC \implies AH \perp E'D \implies$ The point $E'$ is symmetric to $D$ with respect to <missing 3>." Since $E'$ is symmetric to $D$ with respect

2,3,1

wenxinyiyan

Detailed reasoning:

First, let's analyze the solution step by step to understand what is missing.

1. The solution talks about a point diametrically opposite to $S$ through $S'$, and it is mentioned that $AS'$ is the internal angle bisector of $\angle BAC$.
2. Then, it is mentioned that $BS$ and $AS'$ meet at $H$, and $\angle ABS = \varphi$.
3. After some angle calculations, it is shown that $\triangle AHD \sim \triangle BAH$.
4. The solution then claims that to finish the proof, we need to show that $PH = AH$.
5. Some more constructions and angle calculations follow, leading to the conclusion that $B, L, P, D,$ and $G$ are concyclic.
6. Finally, the solution talks about a lemma and uses it to complete the proof.

Now, let's look at the missing parts and try to match them with the given formulae.

<missing 1>: This should be related to the circle $\Omega$ and its properties. Looking at the formulae, <expression 2> talks about $\Omega$ being the circle $BCD$ with $BC$ as the diameter. This seems to fit here as it describes the properties of $\Omega$ that are used in the solution.

<missing 2>: This part is about applying Reim's theorem to $\omega, \Omega$ and lines $CDD$ and another line. Among the formulae, <expression 3> talks about lines $BE'F'$ which could be the other line mentioned in the missing part.

<missing 3>: This should be related to the symmetry of points with respect to a line. Among the formulae, <expression 1> talks about $AH$ being the line of symmetry for points $E'$ and $D$. This seems to fit here as it explains the symmetry property used in the solution.

Answer: 2, 3, 1

2,3,1

zhipu

### Detailed Reasoning

1. **Understanding the Context of the Missing Tags:**
- **<missing 1>**: This is in the context of describing the circle $\Omega$ where $BC$ is the diameter. We need a formula that fits this description.
- **<missing 2>**: This is used in the application of Reim’s theorem, involving the circles $\omega$ and $\Omega$ and the lines $CD$ and another line. We need a formula that describes a relevant line or property.
- **<missing 3>**: This is in the context of symmetry with respect to a line or point. We need a formula that fits this description.

2. **Analyzing the Given Formulae:**
- **<expression 1>**: `AH \implies E' = E \implies F' = F \implies HF = HD.` This seems to be a conclusion or a sequence of implications rather than a descriptive formula for a circle or line.
- **<expression 2>**: `\Omega.` This directly refers to the circle $\Omega$.
- **<expression 3>**: `BE'F'`. This refers to a set of points or a line involving $B$, $E'$, and $F'$.

3. **Matching the Formulae to the Missing Tags:**
- **<missing 1>**: The description "BC is the diameter of <missing 1>" suggests we need something that describes the circle $\Omega$. The most fitting formula here is `<expression 2> \Omega.` because it directly refers to the circle $\Omega$.
- **<missing 2>**: In the context of Reim’s theorem, we need a line or a set of points that are relevant. `<expression 3> BE'F'` seems to fit this context as it involves points on the circle and lines related to $B$, $E'$, and $F'$.
- **<missing 3>**: The context is about symmetry with respect to a line or point. `<expression 1> AH \implies E' = E \implies F' = F \implies HF = HD.` seems to describe a symmetry or equivalence, fitting the context of symmetry.

### Final Answer

Answer: 2, 3, 1

2,3,1

deepseek-v3

<Detailed reasoning>

1. **Identifying <missing 1>:** In the lemma, the sentence "BC is the diameter of <missing 1>" suggests that <missing 1> should be the circle for which BC is the diameter. Given the context, the circle in question is $\Omega$, which is the circle $BCD$. Therefore, <missing 1> should be replaced by $\Omega$. The corresponding expression is <expression 2>.

2. **Identifying <missing 2>:** In the lemma, the sentence "We use Reim’s theorem for $\omega$, $\Omega$ and lines $CDD$ and <missing 2>" suggests that <missing 2> should be the line that is being compared to $CDD$. Given the context, this line is $BE'F'$. Therefore, <missing 2> should be replaced by $BE'F'$. The corresponding expression is <expression 3>.

3. **Identifying <missing 3>:** In the lemma, the sentence "The point $E'$ is symmetric to $D$ with respect to <missing 3>" suggests that <missing 3> should be the line or point with respect to which $E'$ is symmetric to $D$. Given the context, this is $AH$. Therefore, <missing 3> should be replaced by $AH$. The corresponding expression is <expression 1>.

Answer: 2, 3, 1

2,3,1