2017 IMO Problems/Problem 4
Let and
be different points on a circle
such that
is not a diameter. Let
be the tangent line to
at
. Point
is such that
is the midpoint of the line segment
. Point
is chosen on the shorter arc
of
so that the circumcircle
of triangle
intersects
at two distinct points. Let
be the common point of
and
that is closer to
. Line
meets
again at
. Prove that the line
is tangent to
.
Solution
We construct inversion which maps into the circle
and
into
Than we prove that
is tangent to
Quadrungle is cyclic
Quadrungle
is cyclic
We construct circle
centered at
which maps
into
Let
Inversion with respect
swap
and
maps into
Inversion with respect
maps
into
.