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:
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.
-In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.

The solution is:
Let $X$ be the foot from $A$ to $\overline{BC}$ . By definition, $\angle AXM = \angle MPC = 90^{\circ}$ . Thus, $\triangle AXM \sim \triangle MPC$ , and $\triangle BMP \sim \triangle AMQ$ .
-Let $X$ be the foot from $A$ to $\overline{BC}$ . By definition, $\angle AXM = \angle MPC = 90^{\circ}$ . Thus, $\triangle AXM \sim \triangle MPC$ , and $\triangle BMP \sim \triangle AMQ$ .

From this, we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$ , as <missing 1> . Thus, $M$ is also the midpoint of $XQ$ .

Now, $NB = NC$ if $N$ lies on the perpendicular bisector of $\overline{BC}$ . As $N$ lies on the perpendicular bisector of $\overline{XQ}$ , which is also the perpendicular bisector of $\overline{BC}$ (as $M$ is also the midpoint of $XQ$ ), we are done.

The formulae are:
<expression 1> MC=MB

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	<Detailed reasoning> The solution provided requires us to prove that \(NB=NC\), and it does so by establishing that \(N\) lies on the perpendicular bisector of \(\overline{BC}\). This is true if \(M\) is also the midpoint of \(\overline{XQ}\), which in turn is established by showing that \(\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}\). The missing part in the solution, indicated by <missing 1>, should contain the expression that confirms the equality of ratios leading to the conclusion that \(M\) is the midpoint of \(\overline{XQ}\). From the given formulae, the only formula that directly relates to the midpoint of \(\overline{BC}\) is <expression 1> which states \(MC=MB\). This expression is key to establishing the similarity ratios that lead to \(M\) being the midpoint of \(\overline{XQ}\), which in turn implies \(N\) lies on the perpendicular bisector of \(\overline{BC}\). Answer: 1	1
deepseek-r1	The solution uses the fact that M is the midpoint of BC (hence MC=MB) to establish ratios from similar triangles. Specifically, from the similarity of triangles BMP and AMQ, the ratio BM/MA = MP/MQ becomes MC/MA = MP/MQ because BM=MC. Combining this with other ratios leads to MX=MQ, showing M is the midpoint of XQ. The key step relies on MC=MB (expression 1) to equate the ratios. Answer: 1	1
gpto1	Error code: 400 - {'error': {'message': 'Invalid prompt: your prompt was flagged as potentially violating our usage policy. Please try again with a different prompt: https://platform.openai.com/docs/guides/reasoning/advice-on-prompting', 'type': 'invalid_request_error', 'param': None, 'code': 'invalid_prompt'}}	1
gpt-4o-2024-08-06	To solve this problem, we need to fill in the missing formula in the solution with the correct expression from the provided list. Let's break down the solution step by step: 1. The solution begins by establishing that \(\angle AXM = \angle MPC = 90^{\circ}\), which leads to the similarity of triangles \(\triangle AXM \sim \triangle MPC\) and \(\triangle BMP \sim \triangle AMQ\). 2. From the similarity of these triangles, the solution derives the proportion: \[ \frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB} \] This proportion is crucial because it implies a relationship between the segments \(MC\), \(MA\), \(MB\), and \(MQ\). 3. The missing part of the solution is marked as <missing 1>, which is supposed to justify why \(M\) is also the midpoint of \(XQ\). 4. The solution then concludes that \(NB = NC\) if \(N\) lies on the perpendicular bisector of \(\overline{BC}\). Since \(N\) is on the perpendicular bisector of \(\overline{XQ}\), and \(M\) is the midpoint of both \(\overline{BC}\) and \(\overline{XQ}\), the condition is satisfied. Now, let's examine the provided formula: - <expression 1> is \(MC = MB\). This expression is a direct statement that \(M\) is the midpoint of \(\overline{BC}\), which is consistent with the given condition that \(M\) is the midpoint of \(\overline{BC}\). The missing part of the solution is trying to establish that \(M\) is also the midpoint of \(\overline{XQ}\). Since the solution already states that \(M\) is the midpoint of \(\overline{BC}\), and the only provided expression is \(MC = MB\), it seems that the solution is using this fact to argue that \(M\) is also the midpoint of \(\overline{XQ}\). Therefore, the missing expression that justifies \(M\) being the midpoint of \(\overline{XQ}\) is likely relying on the fact that \(MC = MB\). Thus, the missing expression is: Answer: 1	1
DeepSeek-Coder-V2-Lite-Instruct	<Detailed reasoning> First, we need to establish the similarity between the triangles. Given that $\angle AXM = \angle MPC = 90^{\circ}$, we can use the AA similarity criterion to conclude that $\triangle AXM \sim \triangle MPC$. Additionally, since $\triangle BMP \sim \triangle AMQ$, we can use the properties of similar triangles to find relationships between the sides. To find the relationship between the sides, we need to match the given expressions to the missing tags in the solution. The expression that relates $MC$ and $MB$ is: <expression 1> MC=MB This expression matches the relationship we need to use to show that $M$ is the midpoint of $XQ$ and thus the perpendicular bisector of $\overline{BC}$ and $\overline{XQ}$ are the same. Answer: 1	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 solution states that "From this, we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$, as <missing 1>." The formulae given are "<expression 1> MC=MB". Since the missing tag is related to the equality of ratios, and the given formula is a simple equality between two lengths, it seems that the formula is not directly applicable to the missing tag. However, upon closer inspection, we can see that the equality of ratios is actually a consequence of the similarity of triangles, and the given formula is a simple equality between two lengths that can be used to support the similarity of triangles. Therefore, the best guess for the missing tag is the given formula, which is expression 1. Answer: 1	1
Meta-Llama-3.1-70B-Instruct-Turbo	To solve this problem, we need to match the given formulae with the missing tags in the solution. The solution states that $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$, as <missing 1>. We need to find a formula that matches this equation. Looking at the given formulae, we see that <expression 1> is $MC=MB$. This formula is not exactly the same as the equation in the solution, but it does involve the variables $MC$ and $MB$, which are part of the equation. Since there is only one formula given, and it's not a perfect match, we can try to manipulate the formula to see if it can be used to fill in the missing tag. However, upon closer inspection, we can see that the equation in the solution is actually a series of equalities, and <expression 1> $MC=MB$ is one of the equalities. Therefore, it's likely that <expression 1> is the correct match for <missing 1>. Answer: 1	1
Qwen2-72B-Instruct	The solution mentions that "we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$ , as <missing 1> ." This statement is essentially saying that since these ratios are equal, it implies something about the lengths involved. Looking at the expressions provided, expression 1 states that $MC=MB$, which is a direct consequence of the fact that $M$ is the midpoint of $BC$. This equality is used to justify that $M$ is also the midpoint of $XQ$, which is crucial for the argument that $N$ lies on the perpendicular bisector of $BC$ and thus $NB=NC$. Therefore, <missing 1> corresponds to expression 1. Answer: 1	1
baichuan4	To solve this problem, we need to match the missing formulae in the solution with the provided expressions. Let's analyze the context and determine which expression fits where. 1. In the solution, it states that "From this, we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$ , as <missing 1>." This part of the solution is explaining why certain ratios are equal. The missing formula should relate the lengths of the segments mentioned (MC, MA, MB) in a way that justifies these equalities. 2. The only formula provided is <expression 1>: MC=MB. This formula directly relates the lengths of MC and MB, which are mentioned in the solution where <missing 1> is located. 3. Given that the solution mentions ratios involving MC and MB, and the only formula provided also involves MC and MB, it seems reasonable to conclude that <expression 1> is the correct formula to fill in for <missing 1>. 4. There are no other missing tags in the solution, so we do not need to consider any additional expressions. Based on this analysis, the answer is: <Detailed reasoning> The solution refers to a series of equal ratios involving the segments MC, MA, and MB. The provided formula MC=MB supports the equality of ratios by establishing that MC and MB have the same length, which is a necessary condition for the ratios to be equal as stated in the solution. Therefore, <expression 1> correctly fills in <missing 1> in the solution. Answer: 1	1
doubao	<The similarity of the triangles gives the equal ratios. In the given expressions, MC=MB fills the <missing 1>.> Answer: 1	1
gpto1	Error code: 400 - {'error': {'message': 'Invalid prompt: your prompt was flagged as potentially violating our usage policy. Please try again with a different prompt: https://platform.openai.com/docs/guides/reasoning/advice-on-prompting', 'type': 'invalid_request_error', 'param': None, 'code': 'invalid_prompt'}}	1
kimi	<Detailed reasoning> To solve this problem, we need to match the given formulae with the missing tags in the solution. Let's analyze the solution step by step: 1. The solution states that $\triangle AXM \sim \triangle MPC$ and $\triangle BMP \sim \triangle AMQ$. This similarity implies that the ratios of corresponding sides are equal. 2. From the similarity, we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$. This equation contains the missing tag <missing 1>. 3. The solution then states that $M$ is the midpoint of $XQ$, which is derived from the previous equation. 4. To prove that $NB = NC$, we need to show that $N$ lies on the perpendicular bisector of $\overline{BC}$. The solution states that $N$ lies on the perpendicular bisector of $\overline{XQ}$, which is also the perpendicular bisector of $\overline{BC}$ since $M$ is the midpoint of $XQ$. Now, let's look at the given formulae: <expression 1> MC=MB Comparing this formula with the equation in the solution, we can see that it matches the missing part of the equation $\frac{MC}{MA} = \frac{MA}{MB}$. Therefore, expression 1 should fill the <missing 1> tag. Answer: 1	1
minimax	To solve this problem, we need to understand the relationships between the segments and angles in the given configuration. We are given that $\triangle AXM \sim \triangle MPC$ and $\triangle BMP \sim \triangle AMQ$. This similarity implies that corresponding sides are proportional. From the similarity of $\triangle AXM \sim \triangle MPC$, we can write the proportionality of their sides as: $$ \frac{MP}{MX} = \frac{MC}{MA} $$ Similarly, from the similarity of $\triangle BMP \sim \triangle AMQ$, we can write the proportionality of their sides as: $$ \frac{MP}{MQ} = \frac{MA}{MB} $$ Combining these two proportions, we can see that: $$ \frac{MC}{MA} = \frac{MA}{MB} $$ This implies that: $$ MC \cdot MB = MA^2 $$ However, we are looking for a formula that directly relates $MC$ and $MB$. The correct relationship that would allow us to conclude that $NB = NC$ is that $MC = MB$. This is because if $MC = MB$, then $N$, being the midpoint of $AQ$, would lie on the	1
wenxinyiyan	<Detailed reasoning> In the solution, it is mentioned that "From this, we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$ , as <missing 1>." Here, the missing part should refer to the fact that the ratios are equal because of similar triangles. However, the provided formula "<expression 1> MC=MB" does not directly fit into this context, as it merely states that $MC$ is equal to $MB$, which is true by definition since $M$ is the midpoint of $\overline{BC}$. Nevertheless, if we were to fill in the missing part with a formula that justifies the equality of the ratios, it would be based on the similarity of triangles. But since the provided formula does not directly justify the equality of the ratios, there might be a misunderstanding or a mistake in the provided formula. Given the information and the formula provided, there seems to be a mismatch between the expected justification (similarity of triangles leading to equality of ratios) and the formula given (mere equality of two segments). Answer: 1 (based on the assumption that the formula provided is meant to justify the equality of ratios, even though it does not directly do so)	1
zhipu	### Detailed Reasoning 1. Understanding the Problem and Solution: - We are given a geometric problem involving a triangle \(ABC\), various points, and their relationships. - The goal is to prove that \(NB = NC\). 2. Analyzing the Solution: - The solution introduces point \(X\) as the foot from \(A\) to \(\overline{BC}\). - It establishes that \(\triangle AXM \sim \triangle MPC\) and \(\triangle BMP \sim \triangle AMQ\). - From these similarities, certain ratios are derived. 3. Identifying the Missing Formula: - The solution states: \(\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}\) as <missing 1>. - This implies that <missing 1> should be a statement that justifies these equal ratios. 4. Given Formulae: - <expression 1> \(MC = MB\). 5. Matching the Formulae to the Missing Tags: - The given formula \(MC = MB\) directly supports the equality \(\frac{MC}{MA} = \frac{MA}{MB}\) because if \(MC = MB\), then the ratios involving these segments can be equated appropriately. ### Conclusion The missing part in the solution is directly related to the equality \(MC = MB\), which is provided as <expression 1>. Therefore, <expression 1> should fill <missing 1>. ### Final Answer Answer: 1	1
deepseek-v3	<Detailed reasoning> 1. The first missing part in the solution is: "From this, we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$, as <missing 1>." The reasoning here is that since $M$ is the midpoint of $\overline{BC}$, we have $MC = MB$. This is given by the formula <expression 1>: $MC = MB$. Therefore, this is the correct formula to fill in the missing part. Answer: 1	1

模型

模型答案

正确答案

金声玉亮2.0

<Detailed reasoning>

The solution provided requires us to prove that $NB=NC$, and it does so by establishing that $N$ lies on the perpendicular bisector of $\overline{BC}$. This is true if $M$ is also the midpoint of $\overline{XQ}$, which in turn is established by showing that $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$. The missing part in the solution, indicated by <missing 1>, should contain the expression that confirms the equality of ratios leading to the conclusion that $M$ is the midpoint of $\overline{XQ}$.

From the given formulae, the only formula that directly relates to the midpoint of $\overline{BC}$ is <expression 1> which states $MC=MB$. This expression is key to establishing the similarity ratios that lead to $M$ being the midpoint of $\overline{XQ}$, which in turn implies $N$ lies on the perpendicular bisector of $\overline{BC}$.

Answer: 1

deepseek-r1

The solution uses the fact that M is the midpoint of BC (hence MC=MB) to establish ratios from similar triangles. Specifically, from the similarity of triangles BMP and AMQ, the ratio BM/MA = MP/MQ becomes MC/MA = MP/MQ because BM=MC. Combining this with other ratios leads to MX=MQ, showing M is the midpoint of XQ. The key step relies on MC=MB (expression 1) to equate the ratios.

Answer: 1

gpto1

Error code: 400 - {'error': {'message': 'Invalid prompt: your prompt was flagged as potentially violating our usage policy. Please try again with a different prompt: https://platform.openai.com/docs/guides/reasoning/advice-on-prompting', 'type': 'invalid_request_error', 'param': None, 'code': 'invalid_prompt'}}

gpt-4o-2024-08-06

To solve this problem, we need to fill in the missing formula in the solution with the correct expression from the provided list.

Let's break down the solution step by step:

1. The solution begins by establishing that $\angle AXM = \angle MPC = 90^{\circ}$, which leads to the similarity of triangles $\triangle AXM \sim \triangle MPC$ and $\triangle BMP \sim \triangle AMQ$.

2. From the similarity of these triangles, the solution derives the proportion:
\[
\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}
\]
This proportion is crucial because it implies a relationship between the segments $MC$, $MA$, $MB$, and $MQ$.

3. The missing part of the solution is marked as <missing 1>, which is supposed to justify why $M$ is also the midpoint of $XQ$.

4. The solution then concludes that $NB = NC$ if $N$ lies on the perpendicular bisector of $\overline{BC}$. Since $N$ is on the perpendicular bisector of $\overline{XQ}$, and $M$ is the midpoint of both $\overline{BC}$ and $\overline{XQ}$, the condition is satisfied.

Now, let's examine the provided formula:

- <expression 1> is $MC = MB$.

This expression is a direct statement that $M$ is the midpoint of $\overline{BC}$, which is consistent with the given condition that $M$ is the midpoint of $\overline{BC}$.

The missing part of the solution is trying to establish that $M$ is also the midpoint of $\overline{XQ}$. Since the solution already states that $M$ is the midpoint of $\overline{BC}$, and the only provided expression is $MC = MB$, it seems that the solution is using this fact to argue that $M$ is also the midpoint of $\overline{XQ}$.

Therefore, the missing expression that justifies $M$ being the midpoint of $\overline{XQ}$ is likely relying on the fact that $MC = MB$.

Thus, the missing expression is:

Answer: 1

DeepSeek-Coder-V2-Lite-Instruct

<Detailed reasoning>

First, we need to establish the similarity between the triangles. Given that $\angle AXM = \angle MPC = 90^{\circ}$, we can use the AA similarity criterion to conclude that $\triangle AXM \sim \triangle MPC$. Additionally, since $\triangle BMP \sim \triangle AMQ$, we can use the properties of similar triangles to find relationships between the sides.

To find the relationship between the sides, we need to match the given expressions to the missing tags in the solution. The expression that relates $MC$ and $MB$ is:

<expression 1> MC=MB

This expression matches the relationship we need to use to show that $M$ is the midpoint of $XQ$ and thus the perpendicular bisector of $\overline{BC}$ and $\overline{XQ}$ are the same.

Answer:
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 solution states that "From this, we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$, as <missing 1>."

The formulae given are "<expression 1> MC=MB".

Since the missing tag is related to the equality of ratios, and the given formula is a simple equality between two lengths, it seems that the formula is not directly applicable to the missing tag. However, upon closer inspection, we can see that the equality of ratios is actually a consequence of the similarity of triangles, and the given formula is a simple equality between two lengths that can be used to support the similarity of triangles.

Therefore, the best guess for the missing tag is the given formula, which is expression 1.

Answer: 1

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

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

The solution states that $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$, as <missing 1>. We need to find a formula that matches this equation.

Looking at the given formulae, we see that <expression 1> is $MC=MB$. This formula is not exactly the same as the equation in the solution, but it does involve the variables $MC$ and $MB$, which are part of the equation.

Since there is only one formula given, and it's not a perfect match, we can try to manipulate the formula to see if it can be used to fill in the missing tag.

However, upon closer inspection, we can see that the equation in the solution is actually a series of equalities, and <expression 1> $MC=MB$ is one of the equalities. Therefore, it's likely that <expression 1> is the correct match for <missing 1>.

Answer: 1

Qwen2-72B-Instruct

The solution mentions that "we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$ , as <missing 1> ." This statement is essentially saying that since these ratios are equal, it implies something about the lengths involved. Looking at the expressions provided, expression 1 states that $MC=MB$, which is a direct consequence of the fact that $M$ is the midpoint of $BC$. This equality is used to justify that $M$ is also the midpoint of $XQ$, which is crucial for the argument that $N$ lies on the perpendicular bisector of $BC$ and thus $NB=NC$. Therefore, <missing 1> corresponds to expression 1.

Answer: 1

baichuan4

To solve this problem, we need to match the missing formulae in the solution with the provided expressions. Let's analyze the context and determine which expression fits where.

1. In the solution, it states that "From this, we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$ , as <missing 1>." This part of the solution is explaining why certain ratios are equal. The missing formula should relate the lengths of the segments mentioned (MC, MA, MB) in a way that justifies these equalities.

2. The only formula provided is <expression 1>: MC=MB. This formula directly relates the lengths of MC and MB, which are mentioned in the solution where <missing 1> is located.

3. Given that the solution mentions ratios involving MC and MB, and the only formula provided also involves MC and MB, it seems reasonable to conclude that <expression 1> is the correct formula to fill in for <missing 1>.

4. There are no other missing tags in the solution, so we do not need to consider any additional expressions.

Based on this analysis, the answer is:

<Detailed reasoning>
The solution refers to a series of equal ratios involving the segments MC, MA, and MB. The provided formula MC=MB supports the equality of ratios by establishing that MC and MB have the same length, which is a necessary condition for the ratios to be equal as stated in the solution. Therefore, <expression 1> correctly fills in <missing 1> in the solution.

Answer: 1

doubao

<The similarity of the triangles gives the equal ratios. In the given expressions, MC=MB fills the <missing 1>.>
Answer: 1

gpto1

kimi

<Detailed reasoning>

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

1. The solution states that $\triangle AXM \sim \triangle MPC$ and $\triangle BMP \sim \triangle AMQ$. This similarity implies that the ratios of corresponding sides are equal.

2. From the similarity, we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$. This equation contains the missing tag <missing 1>.

3. The solution then states that $M$ is the midpoint of $XQ$, which is derived from the previous equation.

4. To prove that $NB = NC$, we need to show that $N$ lies on the perpendicular bisector of $\overline{BC}$. The solution states that $N$ lies on the perpendicular bisector of $\overline{XQ}$, which is also the perpendicular bisector of $\overline{BC}$ since $M$ is the midpoint of $XQ$.

Now, let's look at the given formulae:

<expression 1> MC=MB

Comparing this formula with the equation in the solution, we can see that it matches the missing part of the equation $\frac{MC}{MA} = \frac{MA}{MB}$. Therefore, expression 1 should fill the <missing 1> tag.

Answer: 1

minimax

To solve this problem, we need to understand the relationships between the segments and angles in the given configuration. We are given that $\triangle AXM \sim \triangle MPC$ and $\triangle BMP \sim \triangle AMQ$. This similarity implies that corresponding sides are proportional.

From the similarity of $\triangle AXM \sim \triangle MPC$, we can write the proportionality of their sides as:
$$
\frac{MP}{MX} = \frac{MC}{MA}
$$

Similarly, from the similarity of $\triangle BMP \sim \triangle AMQ$, we can write the proportionality of their sides as:
$$
\frac{MP}{MQ} = \frac{MA}{MB}
$$

Combining these two proportions, we can see that:
$$
\frac{MC}{MA} = \frac{MA}{MB}
$$

This implies that:
$$
MC \cdot MB = MA^2
$$

However, we are looking for a formula that directly relates $MC$ and $MB$. The correct relationship that would allow us to conclude that $NB = NC$ is that $MC = MB$. This is because if $MC = MB$, then $N$, being the midpoint of $AQ$, would lie on the

wenxinyiyan

<Detailed reasoning>

In the solution, it is mentioned that "From this, we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$ , as <missing 1>." Here, the missing part should refer to the fact that the ratios are equal because of similar triangles. However, the provided formula "<expression 1> MC=MB" does not directly fit into this context, as it merely states that $MC$ is equal to $MB$, which is true by definition since $M$ is the midpoint of $\overline{BC}$.

Nevertheless, if we were to fill in the missing part with a formula that justifies the equality of the ratios, it would be based on the similarity of triangles. But since the provided formula does not directly justify the equality of the ratios, there might be a misunderstanding or a mistake in the provided formula.

Given the information and the formula provided, there seems to be a mismatch between the expected justification (similarity of triangles leading to equality of ratios) and the formula given (mere equality of two segments).

Answer: 1 (based on the assumption that the formula provided is meant to justify the equality of ratios, even though it does not directly do so)

zhipu

### Detailed Reasoning

1. **Understanding the Problem and Solution:**
- We are given a geometric problem involving a triangle $ABC$, various points, and their relationships.
- The goal is to prove that $NB = NC$.

2. **Analyzing the Solution:**
- The solution introduces point $X$ as the foot from $A$ to $\overline{BC}$.
- It establishes that $\triangle AXM \sim \triangle MPC$ and $\triangle BMP \sim \triangle AMQ$.
- From these similarities, certain ratios are derived.

3. **Identifying the Missing Formula:**
- The solution states: $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$ as <missing 1>.
- This implies that <missing 1> should be a statement that justifies these equal ratios.

4. **Given Formulae:**
- <expression 1> $MC = MB$.

5. **Matching the Formulae to the Missing Tags:**
- The given formula $MC = MB$ directly supports the equality $\frac{MC}{MA} = \frac{MA}{MB}$ because if $MC = MB$, then the ratios involving these segments can be equated appropriately.

### Conclusion
The missing part in the solution is directly related to the equality $MC = MB$, which is provided as <expression 1>. Therefore, <expression 1> should fill <missing 1>.

### Final Answer
Answer: 1

deepseek-v3

<Detailed reasoning>

1. The first missing part in the solution is: "From this, we have $\frac{MP}{MX} = \frac{MC}{MA} = \frac{MP}{MQ} = \frac{MA}{MB}$, as <missing 1>."

The reasoning here is that since $M$ is the midpoint of $\overline{BC}$, we have $MC = MB$. This is given by the formula <expression 1>: $MC = MB$. Therefore, this is the correct formula to fill in the missing part.

Answer: 1