# 2001 JBMO Problems/Problem 3

## Problem 3

Let be an equilateral triangle and on the sides and respectively. If (with ) are the interior angle bisectors of the angles of the triangle , prove that the sum of the areas of the triangles and is at most equal with the area of the triangle . When does the equality hold?

## Solution

We have Similarly Thus . We have , and , so Obviously , so it is sufficient to prove that . The cosine rule applied to gives . Hence also . Thus we have .

So , which is .

## Bonus Question

In the above problem, prove that .

- Proposed by

## Solution to Bonus Question

Let be the intersection of and , so is an angle bisector of triangle . Extend line DE to meet BC at H. Let us define

We have and Thus .

It follows that is a cyclic quadrilateral. So we have , and

So , implying that triangle is an isoceles triangle. So we have .

Also, since , and , it follows that: trianlge ~ triangle

Similarly it can be shown that: trianlge ~ triangle .

From trianlge ~ triangle we get: , or

From triangle ~ triangle we get: , or

Since , we get

or

or

Thus