Lor
2023 USAMO
Problem 6
Let ABC be a triangle with incenter 
and excenters 
, 
, 
opposite 
, 
, and 
, respectively. Given an arbitrary point 
on the circumcircle of 
that does not lie on any of the lines 
, 
, or 
, suppose the circumcircles of 
and 
intersect at two distinct points and 
. If 
is the intersection of lines 
and , prove that 
.
Important Ideas in Geometry(From 50 Important Formulas in Geometry)
Cyclic Quadrilaterals
Angle Bisectors
Angle Chasing
Similar Triangles
Power of a Point
Radical Axis
Linearity of Power of a Point
AI.AIa=bc=AIb.AIc
BICIa is cyclic
Suppose extension of AD intersects circumcircle of DIbIc at X
If you draw perpendicular from X to AIb and extend it then it intersects BC at E’
AX=AE’
Suppose extension of AD intersects circumcircle of DIIa at Y
If you draw perpendicular from Y to AIa and extend it then it intersects BC at E’
YN=NE’
Solution