2023 IMO Problems/Problem 2
Problem
Let be an acute-angled triangle with
. Let
be the circumcircle of
. Let
be the midpoint of the arc
of
containing
. The perpendicular from
to
meets
at
and meets
again at
. The line through
parallel to
meets line
at
. Denote the circumcircle of triangle
by
. Let
meet
again at
. Prove that the line tangent to
at
meets line
on the internal angle bisector of
.
Solution
Denote the point diametrically opposite to a point through
is the internal angle bisector of
.
Denote the crosspoint of and
through
To finishing the solution we need only to prove that
Denote
is incenter of
Denote is the orthocenter of
Denote and
are concyclic.
points and
are colinear
is symmetric to
with respect
We use the lemma and complete the proof.
Solutions
https://www.youtube.com/watch?v=JhThDz0H7cI [Video contains solutions to all day 1 problems]