1987 IMO Problems/Problem 2
From AoPSWiki
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
.
See also
| 1987 IMO (Problems) | ||
| Preceded by Problem 1 | 1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |



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