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Tangential quadrilateral collinearity on steroids

Source: 2021 Mexico Center Zone Regional Olympiad, problem 3

January 17, 2022

Mexicogeometrycyclic quadrilateraltangential quadrilateralcollinear

Problem Statement

Let

W,X,Y

and

Z

be points on a circumference

\omega

with center

O

, in that order, such that

WY

is perpendicular to

XZ

;

T

is their intersection.

ABCD

is the convex quadrilateral such that

W,X,Y

and

Z

are the tangency points of

\omega

with segments

AB,BC,CD

and

DA

respectively. The perpendicular lines to

OA

and

OB

through

A

and

B

, respectively, intersect at

P

; the perpendicular lines to

OB

and

OC

through

B

and

C

, respectively, intersect at

Q

, and the perpendicular lines to

OC

and

OD

through

C

and

D

, respectively, intersect at

R

.

O_1

is the circumcenter of triangle

PQR

. Prove that

T,O

and

O_1

are collinear.
Proposed by CDMX

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