Ptolemy’s Theorem
In a cyclic quadrilateral the product of diagonals equals sum of products of opposite sides.
Key idea
In a cyclic quadrilateral the product of diagonals equals sum of products of opposite sides.
In depth
In a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides: AC · BD = AB · CD + AD · BC. It generalises Pythagoras, since a rectangle is cyclic.
Worked example
A rectangle with sides a, b and diagonal d: d·d = a·a + b·b.
Formula
Real-world uses
- Finding diagonal lengths in cyclic quadrilaterals for circular layouts
- Competition geometry
Common mistake
Using it for quadrilaterals that are not cyclic.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Ptolemy’s Theorem?
In a cyclic quadrilateral the product of diagonals equals sum of products of opposite sides. It is a topic at the expert level in Geometry.
What is the formula for Ptolemy’s Theorem?
The key formula is AC·BD = AB·CD + AD·BC.
Where is Ptolemy’s Theorem used in real life?
Real-world examples: Finding diagonal lengths in cyclic quadrilaterals for circular layouts; Competition geometry.
What should I learn before Ptolemy’s Theorem?
Make sure you are comfortable with Menelaus’ Theorem, the previous topic in Geometry.
What is a common mistake with Ptolemy’s Theorem?
Using it for quadrilaterals that are not cyclic.
How can I practise Ptolemy’s Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Ptolemy’s Theorem?
Move on to Law of Sines in Geometry next.
Related expert topics in Geometry
- Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
- Cyclic Quadrilaterals – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
- Cavalieri’s Principle – Solids with equal cross-sectional areas at every height have equal volume.
- Area by Coordinates (Shoelace) – Area of a polygon from vertices.
- Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
- Ellipse – Sum of distances to two foci is constant.