Question

4. In a triangle, let P be the location where two angle bisectors meet. Must P be on the third angle bisector as well? Explain your reasoning.

Answer 1

P must be on the third angle bisector as well

Explanation:

The point P must be on the third angle bisector as well, because it is a theorem in Euclidean geometry that the three interior angle bisectors of a triangle meet in a single point.

This point is called the incenter. In Euclid's Elements, Proposition 4 of Book IV proves that this point is also the center of the inscribed circle of the triangle.

The incenter is defined as the point where the internal angle bisectors of the triangle cross.