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The problem shown in the image asks to analyze a triangle \( ABC \), where the sides are provided: 
- \( AB = 10 \), 
- \( BC = 12 \), 
- \( AC = 14 \).

It mentions points \( P \) and \( Q \) such that the angles \( \angle BPC \) and \( \angle AQC \) are bisectors. The figure also marks an angle \( \beta \) and asks to determine the value of \( \beta \), given that it is part of the triangle’s angle bisector system.

From the problem, it seems like we're working with some geometric properties of angle bisectors and potentially using the law of sines or cosines to calculate \( \beta \). Additionally, the angle bisector theorem or other properties might help solve the problem by linking the proportions of triangle sides.

Would you like me to analyze this problem in more detail and walk through the steps to determine \( \beta \)?

Considere um triângulo ABC de lados medindo AB = 10, BC = 12 e AC = 14. Sendo P e Q pontos de AB e AC tais que BPC e AQC são bissetrizes internas dos ângulos do triângulo ABC, como mostra a figura ao lado, determine o valor de \( \beta \), o ângulo formado entre as bissetrizes BPC e AQC.

Learn how to calculate the angle \( \beta \) formed by the internal bisectors of a triangle ABC with sides AB = 10, BC = 12, and AC = 14. This problem explores key geometric concepts such as the Law of Sines and the Angle Bisector Theorem, suitable for students in Grades 10-12.

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