2019 IMO Problems/Problem 6
Problem
Let be the incenter of acute triangle with . The incircle of is tangent to sides , , and at , , and , respectively. The line through perpendicular to meets again at . Line meets ω again at . The circumcircles of triangles and meet again at . Prove that lines and meet on the line through perpendicular to .
Solution
Step 1
We find an auxiliary point
Let be the antipode of on where is radius
We define
is cyclic
An inversion with respect swap and is the midpoint
Let meets again at We define
Opposite sides of any quadrilateral inscribed in the circle meet on the polar line of the intersection of the diagonals with respect to and meet on the line through perpendicular to The problem is reduced to proving that
Step 2
We find a simplified way to define the point
We define and are bisectrices).
We use the Tangent-Chord Theorem and get
Points and are concyclic.
Step 3
We perform inversion around The straight line maps onto circle We denote this circle We prove that the midpoint of lies on the circle
In the diagram, the configuration under study is transformed using inversion with respect to The images of the points are labeled in the same way as the points themselves. Points and have saved their position. Vertices and have moved to the midpoints of the segments and respectively.
Let be the midpoint
We define
is triangle midline point lies on is parallelogram is midpoint
Step 4
We prove that image of lies on
In the inversion plane the image of point lies on straight line (It is image of circle and on circle
point lies on .
vladimir.shelomovskii@gmail.com, vvsss
See Also
2019 IMO (Problems) • Resources | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Last Problem |
All IMO Problems and Solutions |