2020 IMO Problems/Problem 1
Consider the convex quadrilateral . The point is in the interior of . The following ratio equalities hold: Prove that the following three lines meet in a point: the internal bisectors of angles and and the perpendicular bisector of segment .
Let the perpendicular bisector of meet at point , those two lined meet at at respectively.
As the problem states, denote that . We can express another triple with as well. Since the perpendicular line of meets at point , , which means that points are concyclic since
Similarly, points are concyclic as well, which means five points are concyclic.,
Moreover, since , so the angle bisector if the angle must be the perpendicular line of , so as the angle bisector of , which means those three lines must be concurrent at the circumcenter of the circle containing five points as desired
https://youtu.be/bDHtM1wijbY [Shorter solution, video covers all day 1 problems]
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