MathLabs

Problem 1

Let ABCABC be a triangle, and let DD be a point on side BCBC. A line through DD intersects side ABAB at XX and ray ACAC at YY. The circumcircle of triangle BXDBXD intersects the circumcircle ω\omega of triangle ABCABC again at point Z≠BZ\ne B. The lines ZDZD and ZYZY intersect ω\omega again at VV and WW, respectively. Prove that AB=VWAB=VW.
Step 3 of 3: Compare the final chords
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∠AZB=∠ACB=∠WZV⟹AB=VW\angle AZB=\angle ACB=\angle WZV\Longrightarrow AB=VW
Detailed analysis

Using cyclic quadrilaterals ABZC and ZDCY in turn, angle AZB equals angle ACB and then angle WZV, with the supplementary interpretation if Z lies between W and C. In omega, equal or supplementary inscribed angles subtend equal chords, so AB=VW.