Math Problem Statement

In a triangle ABC, the internal bisectors of ∠B and ∠C meet at P, and the external bisectors of ∠B and ∠C meet at Q. Then the value of ∠BPC + ∠BQC is?

Solution

The problem in the image asks the following:

In triangle ABC\triangle ABC, the internal bisectors of B\angle B and C\angle C meet at PP, and the external bisectors of B\angle B and C\angle C meet at QQ. The value of BPC+BQC\angle BPC + \angle BQC is:

  • (a) 9090^\circ
  • (b) 180180^\circ
  • (c) 270270^\circ
  • (d) 360360^\circ

Solution:

This question involves the concept of angle bisectors and properties of angles in a triangle. Here’s how to approach it:

  1. Angle Bisectors:

    • The internal angle bisectors of a triangle meet at a point known as the incenter (P in this case).
    • The external angle bisectors meet at a point known as the excenter (Q in this case).
  2. Summing the Angles:

    • BPC\angle BPC is the angle formed at the incenter by the internal bisectors.
    • BQC\angle BQC is the angle formed at the excenter by the external bisectors.

A known geometric result states that the sum of BPC\angle BPC (formed by the internal bisectors) and BQC\angle BQC (formed by the external bisectors) in any triangle is always 180180^\circ.

Conclusion:

Therefore, the value of BPC+BQC\angle BPC + \angle BQC is 180\boxed{180^\circ}.

Do you want further clarification or more details on this solution?

Here are 5 related questions you might want to explore:

  1. What is the difference between an incenter and an excenter of a triangle?
  2. How do you construct the internal and external angle bisectors in a triangle?
  3. What is the significance of the incenter in a triangle's properties?
  4. Can the sum of these angles BPC+BQC\angle BPC + \angle BQC ever be greater than 180180^\circ?
  5. What are the other centers of a triangle, and how are they different?

Tip: In triangle geometry, understanding key points like the centroid, incenter, circumcenter, and orthocenter, as well as their properties, can simplify many complex problems!

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Math Problem Analysis

Mathematical Concepts

Triangle Geometry
Angle Bisectors
Incenter and Excenter

Formulas

Sum of angles formed by internal and external bisectors in a triangle

Theorems

Theorem stating that the sum of ∠BPC and ∠BQC in any triangle is 180°

Suitable Grade Level

Grades 9-12