# 2000 APMO Problems/Problem 3

## Statement

Let be a triangle. Let and be the points in which the median and the angle bisector, respectively, at meet the side . Let and be the points in which the perpendicular at to meets and , respectively, and the point in which the perpendicular at to meets produced. Prove that is perpendicular to .

## Solution

The problem can be solved by using coordinate geometry.

Let be the coordinate of , and be the equation of straight line of and . Let , , be the coordinate of , and .

Now the coordinates of and are and respectively. If the coordinate of is , then so .

Let be the coordinate of . .

and .

Clearly, . Hence, is perpendicular to . The proof is done.