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EF goes through the points of tangency of the incircle to AB and AC

Source: Dutch BxMO TST 2016 p3

August 2, 2019
geometryincirclecollinearright triangle

Problem Statement

Let △ABC\vartriangle ABC be a right-angled triangle with ∠A=90o\angle A = 90^o and circumcircle Γ\Gamma. The inscribed circle is tangent to BCBC in point DD. Let EE be the midpoint of the arc ABAB of Γ\Gamma not containing CC and let FF be the midpoint of the arc ACAC of Γ\Gamma not containing BB. (a) Prove that △ABC∼△DEF\vartriangle ABC \sim \vartriangle DEF. (b) Prove that EFEF goes through the points of tangency of the incircle to ABAB and ACAC.
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