1996 USAMO Problems/Problem 3
Problem
Let be a triangle. Prove that there is a line
(in the plane of triangle
) such that the intersection of the interior of triangle
and the interior of its reflection
in
has area more than
the area of triangle
.
Solution
Let the triangle be . Assume
is the largest angle. Let
be the altitude. Assume
, so that
. If
, then reflect in
. If
is the reflection of
, then
and the intersection of the two triangles is just
. But
, so
has more than
the area of
.
If , then reflect in the angle bisector of
. The reflection of
is a point on the segment
and not
. (It lies on the line
because we are reflecting in the angle bisector.
because
. Finally,
because we assumed
does not exceed
). The intersection is at least
. But
.
See Also
1996 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
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