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
See also
2001 JBMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |