2001 IMO Shortlist Problems/G6
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 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
Therefore, if , necessarily .