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concyclic wanted, incircle, circumcircle, midline related in a right triangle

Source: Ukrainian Geometry Olympiad 2017 X p4

December 12, 2018

geometrycircumcircleincirclemidlineConcyclic

Problem Statement

In the right triangle

ABC

with hypotenuse

AB

, the incircle touches

BC

and

AC

at points

{{A}_{1}}

and

{{B}_{1}}

respectively. The straight line containing the midline of

\Delta ABC

, parallel to

AB

, intersects its circumcircle at points

P

and

T

. Prove that points

P,T,{{A}_{1}}

and

{{B}_{1}}

lie on one circle.