Difference between revisions of "2001 JBMO Problems/Problem 3"
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Revision as of 23:58, 8 January 2023
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 |