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SAMO Problem 5: Cyclic quadrilateral through points of two isosceles triangles

Source: South African Mathematics Olympiad 2020, Problem 5

January 17, 2021

geometrycyclic quadrilateralcircumcircle

Problem Statement

Let

ABC

be a triangle, and let

T

be a point on the extension of

AB

beyond

B

, and

U

a point on the extension of

AC

beyond

C

, such that

BT = CU

. Moreover, let

R

and

S

be points on the extensions of

AB

and

AC

beyond

A

such that

AS = AT

and

AR = AU

. Prove that

R

S

T

U

lie on a circle whose centre lies on the circumcircle of

ABC