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A right angled triangle with AC=2AB

Source: Iran Team selection test 2024 - P2

May 19, 2024
geometry

Problem Statement

For a right angled triangle △ABC\triangle ABC with ∠A=90\angle A=90 we have AC=2ABAC=2AB. Point MM is the midpoint of side BCBC and II is incenter of triangle △ABC\triangle ABC. The line passing trough MM and perpendicular to BIBI intersect with lines BIBI and ACAC at points HH and KK respectively. If the semi-line IKIK cuts circumcircle of triangle △ABC\triangle ABC at FF and SS be the second intersection point of line FHFH with circumcircle of triangle △ABC\triangle ABC , then prove that SMSM is tangent to the incircle of triangle △ABC\triangle ABC.
Proposed by Mahdi Etesami Fard
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