Ptolemy's Theorem
Ptolemy's Theorem gives a relationship between the side lengths and the diagonals of a cyclic quadrilateral; it is the equality case of Ptolemy's Inequality. Ptolemy's Theorem frequently shows up as an intermediate step in problems involving inscribed figures.
Contents
Statement
Given a cyclic quadrilateral with side lengths and diagonals :
Proof
Given cyclic quadrilateral extend to such that
Since quadrilateral is cyclic, However, is also supplementary to so . Hence, by AA similarity and
Now, note that (subtend the same arc) and so This yields
However, Substituting in our expressions for and Multiplying by yields .
Problems
2004 AMC 10B Problem 24
In triangle we have , , . Point is on the circumscribed circle of the triangle so that bisects angle . What is the value of ?
Solution: Set 's length as . 's length must also be since and intercept arcs of equal length(because ). Using Ptolemy's Theorem, . The ratio is
Equilateral Triangle Identity
Let be an equilateral triangle. Let be a point on minor arc of its circumcircle. Prove that .
Solution: Draw , , . By Ptolemy's Theorem applied to quadrilateral , we know that . Since , we divide both sides of the last equation by to get the result: .
Regular Heptagon Identity
In a regular heptagon , prove that: .
Solution: Let be the regular heptagon. Consider the quadrilateral . If , , and represent the lengths of the side, the short diagonal, and the long diagonal respectively, then the lengths of the sides of are , , and ; the diagonals of are and , respectively.
Now, Ptolemy's Theorem states that , which is equivalent to upon division by .
1991 AIME Problems/Problem 14
A hexagon is inscribed in a circle. Five of the sides have length and the sixth, denoted by , has length . Find the sum of the lengths of the three diagonals that can be drawn from .
Cyclic Hexagon
A hexagon with sides of lengths 2, 2, 7, 7, 11, and 11 is inscribed in a circle. Find the diameter of the circle.
Solution: Consider half of the circle, with the quadrilateral , being the diameter. , , and . Construct diagonals and . Notice that these diagonals form right triangles. You get the following system of equations:
(Ptolemy's Theorem)
Solving gives