Track / Stage 5 / 152 of 400 #752 of 1964

Problem 752

AIME late

Geometry Difficulty 5.4 Prove it

56. Construct an isosceles triangle given the height and median drawn to the lateral side.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

56. Let the adjacent sides of the parallelogram be denoted by $a$ and $b$ , and the vertices by $A, B, C, D$ ; thus, $A D=B C=a, A B=C D=b$ . Let the point inside the parallelogram be denoted by $O$ , the distance between $A D$ and $B C$ by $h_{1}$ , and the distance between $A B$ and $C D$ by $h_{2}$ . Then we have

$S_{\triangle B O C}+S_{\triangle A O D}=\frac{1}{2} a h_{1} ; S_{\triangle A O B}+S_{\triangle C O D}=\frac{1}{2} b h_{2}, \text { but } a h_{1}=b h_{2} .$

Therefore, $S_{\triangle B O C}+S_{\triangle A O D}=S_{\triangle A O B}+S_{\triangle C O D}$ .

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