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Must the triangle be isosceles?

Source: 42nd International Tournament of Towns, Junior O-Level P2, Spring 2021

February 18, 2023
geometryTournament of Towns

Problem Statement

Let AXAX and BZBZ be altitudes of the triangle ABCABC. Let AYAY and BTBT be its angle bisectors. It is given that angles XAYXAY and ZBTZBT are equal. Does this necessarily imply that ABCABC is isosceles?
The Jury
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