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equal circumcircles wanted, 3 more circles related

Source: 2021 Greek Junior MO p4 (served as Greek JBMO TST p4 since the latter didn't take place)

July 3, 2021
geometryequal circles

Problem Statement

Given a triangleABCABC with AB<BC<ACAB<BC<AC inscribed in circle (c)(c). The circle c(A,AB)c(A,AB) (with center AA and radius ABAB) interects the line BCBC at point DD and the circle (c)(c) at point HH. The circle c(A,AC)c(A,AC) (with center AA and radius ACAC) interects the line BCBC at point ZZ and the circle (c)(c) at point EE. Lines ZHZH and EDED intersect at point TT. Prove that the circumscribed circles of triangles TDZTDZ and TEHTEH are equal.
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