Quick answer
Ptolemy's theorem says that in a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides.
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Ptolemy's theorem is a powerful result about quadrilaterals whose four vertices lie on one circle. Such a shape is called a cyclic quadrilateral, and its side lengths and diagonal lengths are connected in a surprisingly simple way. The theorem matters because it lets you find an unknown side or diagonal when enough other lengths are known.
It also links geometry, trigonometry, and many circle problems in a compact formula.
For a cyclic quadrilateral ABCD, Ptolemy's theorem says that the product of the diagonals equals the sum of the products of opposite sides. In symbols, AC × BD = AB × CD + BC × AD. The condition that all four points lie on the same circle is essential, because the formula is not true for most quadrilaterals.
A common use is to solve for a missing diagonal after measuring or calculating the four sides and the other diagonal.
Understanding Geometry: Ptolemy's Theorem
The important step is proving that the four vertices really share one circle. A diagram can look circular even when it is not. One reliable test uses opposite interior angles.
In a cyclic quadrilateral, those two angles add to one hundred eighty degrees. The reverse is useful too. If a quadrilateral has opposite angles that add to one hundred eighty degrees, it can be drawn on a circle.
This angle test often appears before any length calculation. It tells you whether Ptolemy's theorem is allowed.
Students should label the vertices in their order around the boundary. A mixed-up order can turn side lengths into diagonals and produce a convincing but wrong calculation.
The theorem has a close connection to trigonometry. A chord length in a circle depends on the sine of the angle that sees the chord. Angles standing on the same chord have matching relationships, which is why the lengths in a cyclic quadrilateral fit together so neatly.
One proof draws a point on a diagonal and chooses it so that two smaller triangles have equal angles. Similar triangles then create ratios of lengths. After multiplying those ratios and adding them, the required length relationship appears.
Another proof uses the sine rule in triangles formed by a diagonal. These proofs show that the result is not a rule to memorize without reason. The common circle forces the angle relationships that make the length products work.
Ptolemy's theorem can be used as a check as well as a solving tool. Suppose four side lengths and two diagonal lengths are given for a shape claimed to be cyclic. Calculate the two sides of the theorem separately.
If they do not match, at least one measurement, label, or assumption is wrong. In construction problems, this can reveal that a point was placed slightly off the intended circle. In coordinate geometry, you may first use distance calculations to find side lengths.
The theorem can then reduce a long diagonal calculation to ordinary arithmetic. It is especially helpful when a diagram contains many triangles but only one unknown segment.
A rectangle gives a familiar example of the deeper pattern. Its angles guarantee that it is cyclic, and its two diagonals have equal length. Applying Ptolemy to that shape leads to the same length rule found from the right triangles inside it.
This connection explains why circle geometry can reproduce a result usually taught in a right triangle unit. More advanced problems use regular polygons, since every vertex of a regular polygon lies on one circle. Care is needed with units and multiplication.
Each term in the theorem has units of length squared, not length. That quick unit check helps catch mistakes such as adding a single side length to a product of two lengths. Keep exact values, such as square roots, until the final step when possible, since early rounding can make an equality appear false.
Key Facts
- Ptolemy's theorem applies only to cyclic quadrilaterals.
- Main formula: AC × BD = AB × CD + BC × AD.
- The diagonals are AC and BD when the vertices are named in order A, B, C, D around the circle.
- Opposite side products are AB × CD and BC × AD.
- If AC is unknown, then AC = (AB × CD + BC × AD) / BD.
- Every rectangle is cyclic, so for a rectangle with sides a and b and diagonal d, Ptolemy gives d² = a² + b².
Vocabulary
- Cyclic quadrilateral
- A quadrilateral whose four vertices all lie on the same circle.
- Diagonal
- A line segment that connects two nonadjacent vertices of a polygon.
- Opposite sides
- Two sides of a quadrilateral that do not share a vertex.
- Circumcircle
- A circle that passes through every vertex of a polygon.
- Ptolemy's theorem
- A theorem stating that in a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides.
Common Mistakes to Avoid
- Using Ptolemy's theorem on a noncyclic quadrilateral, which is wrong because the theorem requires all four vertices to lie on one circle.
- Multiplying adjacent sides instead of opposite sides, which changes the relationship and does not match AC × BD = AB × CD + BC × AD.
- Labeling the diagonals incorrectly, which leads to using side lengths in place of AC or BD and gives an invalid equation.
- Forgetting to divide when solving for a missing diagonal, which is wrong because AC × BD must be isolated algebraically, such as AC = (AB × CD + BC × AD) / BD.
Practice Questions
- 1 A cyclic quadrilateral ABCD has AB = 5, BC = 7, CD = 8, AD = 6, and BD = 10. Use Ptolemy's theorem to find AC.
- 2 In a cyclic quadrilateral, AB = 4, BC = 9, CD = 6, AD = 5, and AC = 7. Find BD.
- 3 A quadrilateral has side lengths and diagonals that satisfy AC × BD = AB × CD + BC × AD. Explain why this equation alone does not prove the quadrilateral is cyclic unless the vertices are in the correct order and the geometric conditions are verified.