1987 IMO Problems/Problem 2
Problem
In an acute-angled triangle the interior bisector of the angle
intersects
at
and intersects the circumcircle of
again at
. From point
perpendiculars are drawn to
and
, the feet of these perpendiculars being
and
respectively. Prove that the quadrilateral
and the triangle
have equal areas.
Solution
We are to prove that or equivilently,
. Thus, we are to prove that
. It is clear that since
, the segments
and
are equal. Thus, we have
since cyclic quadrilateral
gives
. Thus, we are to prove that
From the fact that and that
is iscoceles, we find that
. So, we have
. So we are to prove that
We have ,
,
,
,
, and so we are to prove that
We shall show that this is true: Let the altitude from touch
at
. Then it is obvious that
and
and thus
.
Thus we have proven that .
1987 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |