MathDB

In a scalene triangle

ABC,

I

is the incenter and

O

is the circumcenter. The line

IO

intersects the lines

BC,CA,AB

at points

D,E,F

respectively. Let

A_1

be the intersection of

BE

and

CF

. The points

B_1

and

C_1

are defined similarly. The incircle of

ABC

is tangent to sides

BC,CA,AB

at points

X,Y,Z

respectively. Let the lines

XA_1, YB_1

and

ZC_1

intersect

IO

at points

A_2,B_2,C_2

respectively. Prove that the circles with diameters

AA_2,BB_2

and

CC_2

have a common point.

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