MathDB

An isosceles triangle

ABC

(

AB = BC

) is given. Circles

\omega_1

and

\omega_2

with centres

O_1

and

O_2

lie in the angle

ABC

and touch the sides

AB

and

CB

A

and

C

respectively, and touch each other externally at point

X

. The side

AC

meets the circles again at points

Y

and

Z

O

is the circumcenter of the triangle

XYZ

. Lines

O_2 O

and

O_1 O

intersect lines

AB

and

BC

at points

C_1

and

A_1

respectively. Prove that

B

is the circumcentre of the triangle

A_1 OC_1

Problem Statement