Track / Stage 5 / 212 of 400 #812 of 2000

Problem 812

AIME late

Geometry Difficulty 5.6 Prove it

71. (8th grade) Prove that the square of the bisector of the angle at the vertex of a triangle is equal to the difference between the product of the lateral sides and the product of the segments of the base.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

71. Describe a circle around triangle $A B C$ and extend the bisector of angle $B$ until it intersects the circle at point $K$ . Using the similarity of triangles $A B K$ and $B F C$ ( $F$ - the intersection of $A C$ and $B K$ ), it is easy to prove the required.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.