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CU, PI concurrent with circumcircle, incircle, 1 more circumcircle

Source: 2022 Austrian Regional Competition For Advanced Students p3

October 4, 2022

geometryincenterconcurrencyconcurrent

Problem Statement

Let

ABC

denote a triangle with

AC\ne BC

. Let

I

and

U

denote the incenter and circumcenter of the triangle

ABC

, respectively. The incircle touches

BC

and

AC

in the points

D

and E, respectively. The circumcircles of the triangles

ABC

and

CDE

intersect in the two points

C

and

P

. Prove that the common point

S

of the lines

CU

and

P I

lies on the circumcircle of the triangle

ABC

.
(Karl Czakler)