Question

XYZ is an equilateral triangle, inscribed in a circle. P is a point on the arc YZ such that X and P are on opposite sides of the chord YZ. Which of the following MUST always be true?

Accepted Answer

D. Quadrilateral $XYPZ$ is cyclic. By Ptolemy's Theorem on cyclic quadrilateral $XYPZ$: $XP \cdot YZ = YP \cdot XZ + PZ \cdot XY$. Since $	riangle XYZ$ is equilateral, $XY = YZ = XZ = s$. $XP \cdot s = YP \cdot s + PZ \cdot s \implies XP = YP + PZ$.

Ptolemy's Theorem in Equilateral Triangle

Explanation

Practise this under exam conditions