MathGoldMedalist

Home Toolkit Problems Competitions About

Sign In Register

Home Toolkit Problems Competitions About

Home
›
Toolkit
›
Ptolemy's Theorem

Ptolemy's Theorem

Lesson · Intermediate

Geometry: Plane Geometry

Cyclic quadrilateral ABCD with diagonals AC and BD

A quadrilateral $ABCD$ is cyclic if and only if:

AB\cdot CD+AD\cdot BC=AC\cdot BD

A regular pentagon has side length 1 and diagonal length $d$ . Find $d$ .

Regular pentagon ABCDE with diagonals AD, AC, and DB

Choose $A,B,C,$ and $D$ . Ptolemy gives

d^2=1+d

Therefore,

d^2-d-1=0

Since $d>0$ ,

\boxed{d=\frac{1+\sqrt5}{2}}

AMC 12A 2024 (Problem 19)
AMC 12B 2025 (Problem 25)

← Back to Toolkit

Home Toolkit Problems Competitions About

Stay Connected

Sign up for updates and new content.

Email address

© 2026 MathGoldMedalist. All rights reserved.