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
.
Video solution
https://youtu.be/bDHtM1wijbY [Shorter solution, video covers all day 1 problems]
Short Video solution(中文解说)in Chinese and subtitle in English
solution 1
Let the perpendicular bisector of meet at point
, those two lines 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 and "Shen Kislay kai" ~ edits by Pearl2008
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
vladimir.shelomovskii@gmail.com, vvsss
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 |