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segments of feet of altitudes // segment of feet of angle biectors, isosceles ?

Source: 2014 Sharygin Geometry Olympiad Final Round 8.2

August 3, 2018
geometryisosceles

Problem Statement

Let AHaAH_a and BHbBH_b be altitudes, ALaAL_a and BLbBL_b be angle bisectors of a triangle ABCABC. It is known that HaHb//LaLbH_aH_b // L_aL_b. Is it necessarily true that AC=BCAC = BC?
(B. Frenkin)
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