# 2001 IMO Shortlist Problems/G6

## Problem

Let be a triangle and an exterior point in the plane of the triangle. Suppose the lines , , meet the sides , , (or extensions thereof) in , , , respectively. Suppose further that the areas of triangles , , are all equal. Prove that each of these areas is equal to the area of triangle itself.

## Solution

Solution 1 by Mewto55555:

We use barycentric coordinates.

So is , is , is , and is , with .

Now, the equation of line is just the line , is just , and is .

Also, is just , is , and is .

Thus, the coordinates of is . Similarly, is at and is at

Now, the ratio to is just

The other ratios are similarly and

Since , we have and we want to show that .

Thus, we have .

Since none of (else would be on one of the sides of ):

.

We know . Substuting:

.

From the first and third, we get that

Now consider first and second;

Subbing back in :

which rearranges to

If , then , so is in the triangle (as all of ) contradiction.

Thus, we have

So,

Thus,

Therefore, if , necessarily .