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
Solution 2 (Three perpendicular bisectors)
The essence of the proof is the replacement of the bisectors of angles by the perpendicular bisectors of the sides of the cyclic pentagon.
Let be the circumcenter of is the perpendicular bisector of and point lies on Then
is cyclic. the bisector of the is the perpendicular bisector of the side of the cyclic that passes through the center
A similar reasoning can be done for the perpendicular bisector of
https://youtu.be/bDHtM1wijbY [Shorter solution, video covers all day 1 problems]
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