In a circle with center O, chord AB equals chord AC. Chord AD cuts BC in E. If AC = 12 and AE = 8, then what is the length of AD.
2026-03-25 17:17:22.1774459042
Cyclic Quadrilateral Property Problem
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Draw the perpendicular bisector of BC. It will go through O and A. Let it cut BC at X.
Reflect AED about AOX to get AE’D’ where E’ and D’ will be on BC and on the circle respectively.
Use Pythagoras theorem on AXC and on AXE to get $12^2 – 8^2 = XC^2 – XE^2$
The RHS $= (XC + XE)[XC – XE] = (CE’)[BE’]$
By power of a point, $ (CE’)[E’B] = AE’ \times E'D’ = 8 \times ED’ = 8 \times ED$. Result follows