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OK _|_ TM wanted, starting with isosceles trapezoid

Source: 2020 Balkan MO shortlist G4

September 14, 2021

perpendiculartrapezoidgeometry

Problem Statement

Let

MAZN

be an isosceles trapezium inscribed in a circle

(c)

with centre

O

. Assume that

MN

is a diameter of

(c)

and let

B

be the midpoint of

AZ

. Let

(\epsilon)

be the perpendicular line on

AZ

passing through

A

. Let

C

be a point on

(\epsilon)

, let

E

be the point of intersection of

CB

with

(c)

and assume that

AE

is perpendicular to

CB

. Let

D

be the point of intersection of

CZ

with

(c)

and let

F

be the antidiametric point of

D

on

(c)

. Let

P

be the point of intersection of

FE

and

CZ

. Assume that the tangents of

(c)

at the points

M

and

Z

meet the lines

AZ

and

PA

at the points

K

and

T

respectively. Prove that

OK

is perpendicular to

TM

.
Theoklitos Parayiou, Cyprus

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