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2 lines concurrent with angle bisector, bisectors, midpoint, right triangle

Source: 2021 Sharygin Finals grades X-XI p1

August 1, 2021
geometryconcurrencyconcurrentright triangle

Problem Statement

.Let CHCH be an altitude of right-angled triangle ABCABC (∠C=90o\angle C = 90^o), HA1HA_1, HB1HB_1 be the bisectors of angles CHBCHB, AHCAHC respectively, and E,FE, F be the midpoints of HB1HB_1 and HA1HA_1 respectively. Prove that the lines AEAE and BFBF meet on the bisector of angle ACBACB.
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