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IT=IA wanted, incircle, circumcircle related (2016 Kyiv City MO 11.4.1)

Source:

September 6, 2020
geometryequal segmentsincirclecircumcircle

Problem Statement

In the triangle ABCABC the angle bisector ADAD is drawn, EE is the point of tangency of the inscribed circle to the side BCBC, II is the center of the inscribed circle ΔABC\Delta ABC. The point A1{{A} _ {1}} on the circumscribed circle ΔABC\Delta ABC is such that AA1∣∣BCA {{A} _ {1}} || BC. Denote by TT - the second point of intersection of the line EA1E {{A} _ {1}} and the circumscribed circle ΔAED\Delta AED. Prove that IT=IAIT = IA.
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