1994 USAMO Problems/Problem 3
Problem
A convex hexagon is inscribed in a circle such that
and diagonals
, and
are concurrent. Let
be the intersection of
and
. Prove that
.
Solution
Let the diagonals ,
,
meet at
.
First, let's show that the triangles and
are similar.
because
,
,
and
all lie on the circle, and
.
because
, and
,
,
,
and
all lie on the circle. Then,
Therefore, and
are similar, so
.
Next, let's show that and
are similar.
because
,
,
and
all lie on the circle, and
.
because
,
,
and
all lie on the circle.
because
, and
,
,
,
and
all lie on the circle. Then,
Therefore, and
are similar, so
.
Lastly, let's show that and
are similar.
Because and
,
,
and
all lie on the circle,
is parallel to
. So,
and
are similar, and
.
Putting it all together, .