2020 IMO Problems/Problem 1
Contents
[hide]Problem
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
.
solution 1
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
~ bluesoul
Video solution
https://youtu.be/bDHtM1wijbY [Shorter solution, video covers all day 1 problems]
See Also
2020 IMO (Problems) • Resources | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |