MathDB

Let

\triangle ABC

be a right-angled triangle with

\angle BAC = 90^{\circ}

and let

E

be the foot of the perpendicular from

A

BC

. Let

Z \ne A

be a point on the line

AB

with

AB = BZ

. Let

(c)

be the circumcircle of the triangle

\triangle AEZ

. Let

D

be the second point of intersection of

(c)

with

ZC

and let

F

be the antidiametric point of

D

with respect to

(c)

. Let

P

be the point of intersection of the lines

FE

and

CZ

. If the tangent to

(c)

Z

meets

PA

T

, prove that the points

T

E

B

Z

are concyclic.
Proposed by Theoklitos Parayiou, Cyprus

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