1985 IMO Problems/Problem 5
Problem
A circle with center passes through the vertices
and
of the triangle
and intersects the segments
and
again at distinct points
and
respectively. Let
be the point of intersection of the circumcircles of triangles
and
(apart from
). Prove that
.
Solution
is the Miquel Point of quadrilateral
, so there is a spiral similarity centered at
that takes
to
. Let
be the midpoint of
and
be the midpoint of
. Thus the spiral similarity must also send
to
and so
is cyclic.
is also cyclic with diameter
and thus
must lie on the same circumcircle as
,
, and
so
.