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Iran TST P8

Source: Iranian TST 2022 problem 8

April 2, 2022
geometrytangentradical axis

Problem Statement

In triangle ABCABC, with AB<ACAB<AC, II is the incenter, EE is the intersection of AA-excircle and BCBC. Point FF lies on the external angle bisector of BACBAC such that EE and FF lieas on the same side of the line AIAI and ∠AIF=∠AEB\angle AIF=\angle AEB. Point QQ lies on BCBC such that ∠AIQ=90\angle AIQ=90. Circle ωb\omega_b is tangent to FQFQ and ABAB at BB, circle ωc\omega_c is tangent to FQFQ and ACAC at CC and both circles pass through the inside of triangle ABCABC. if MM is the Midpoint od the arc BCBC, which does not contain AA, prove that MM lies on the radical axis of ωb\omega_b and ωc\omega_c.
Proposed by Amirmahdi Mohseni
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