2019 IMO Problems/Problem 6
Problem
Let be the incenter of acute triangle
with
. The incircle
of
is tangent to sides
,
, and
at
,
, and
, respectively. The line through
perpendicular to
meets
again at
. Line
meets ω again at
. The circumcircles of triangles
and
meet again at
.
Prove that lines
and
meet on the line through
perpendicular to
.
Solution
Step 1
We find an auxiliary point
Let be the antipode of
on
where
is radius
We define
cyclic
an inversion with respect
swap
and
is the midpoint
Let
meets
again at S (other than D). We define
Opposite sides of any quadrilateral inscribed in the circle
meet on the polar line of the intersection of the diagonals with respect to
and
meet on the line through
perpendicular to
The problem is reduced to proving that