Problem 1

Let

ABC

be a triangle, and let

D

be a point on side

BC

. A line through

D

intersects side

AB

X

and ray

AC

Y

. The circumcircle of triangle

BXD

intersects the circumcircle

\omega

of triangle

ABC

again at point

Z\ne B

. The lines

ZD

and

ZY

intersect

\omega

again at

V

and

W

, respectively. Prove that

AB=VW

Step 1 of 3: Create the first angle chain

Previous step Next step

B,X,D,Z\text{ cyclic},\quad A,B,C,Z,V\text{ cyclic}

Detailed analysis

The points B,X,D,Z are concyclic by construction, and A,B,C,Z,V lie on omega. Since D,Y,X are collinear, cyclic angles give angle ZDY equal to angle ZBA, while A,B,C,Z being cyclic gives angle ZBA equal to angle ZCY.