Ptolemy's Theorem and Its Applications

In the Chinese book: 托勒密定理及其应用

Opposite-side products add up to ≥ the diagonal product; equal iff concyclic.

Try this

Drag $D$ anywhere. Compare the two bars: $A B \cdot C D + A D \cdot B C$ and $A C \cdot B D$ . Does the bottom one ever get longer than the top one?
Click “Snap D onto the circle”. What does “Excess” become?
Keep dragging $D$ along the circle until it passes $A$ and sits on the arc between $A$ and $B$ . $D$ is still on the circle. Does equality still hold?
Imagine $A B C D$ is a rectangle (every rectangle is inscribed in a circle). Then the six segments have only three different lengths. Which famous theorem does the equation turn into?

Ptolemy's theorem: in a cyclic quadrilateral $A B C D$ ,

A B \cdot C D + A D \cdot B C = A C \cdot B D

For any four points in the plane, the left side is at least the right side (Ptolemy's inequality). Equality holds if and only if the four points are concyclic and lie in the order $A$ , $B$ , $C$ , $D$ .

A rectangle has equal opposite sides and equal diagonals, so the theorem becomes $A B^{2} + B C^{2} = A C^{2}$ : the Pythagorean theorem is a special case of Ptolemy's theorem.

Predict first

Equilateral triangle $A B C$ is inscribed in a circle, and point $P$ is on the minor arc $B C$ with $P B = 3$ and $P C = 5$ . What is $P A$ ?

Choose an answer first; then you can see the result.

Why

To prove “a sum of products equals a product”, cut the diagonal $B D$ into two pieces and let each piece take care of one product.

Proof of Ptolemy's theorem

Step 1 / 6

What to prove
$A B C D$ is inscribed in a circle. We want to prove that the sum of the products of the two pairs of opposite sides equals the product of the diagonals.

$A B \cdot C D + A D \cdot B C = A C \cdot B D$
Choose a point E on BD
First pair of similar triangles
Second pair of similar triangles
Add them up
When the points are not concyclic

If D is not on the circle

Break it

Counterexample lab: remove one condition

Conclusion: $A B \cdot C D + A D \cdot B C = A C \cdot B D$

$D$ is on the circle: the two bars are the same length.

Practice

Exercise 1Basic

In isosceles trapezoid $A B C D$ , $A D ∥ B C$ , $A D = 7$ , $B C = 15$ and $A B = C D = 8$ . Find the length of diagonal $A C$ . (An isosceles trapezoid has a circumcircle.)

Exercise 2Basic

Equilateral triangle $A B C$ is inscribed in a circle, and point $P$ is on the minor arc $B C$ with $P A = 10$ and $P B = 4$ . Find $P C$ .

Exercise 3Advanced

Quadrilateral $A B C D$ is inscribed in a circle, $A C$ is a diameter, $A C = 25$ , $A B = 7$ and $A D = 15$ . Find the length of $B D$ .

Exercise 4Challenge

A regular pentagon $A B C D E$ has side $1$ and diagonals of length $x$ . Use Ptolemy's theorem to prove that $x^{2} = x + 1$ , and find $x$ .

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Exhibit

Pairing up adjacent sides

In cyclic quadrilateral

A B C D

A B \cdot B C + C D \cdot D A = A C \cdot B D

What went wrong? Think first, then open

Diagnosis Ptolemy's theorem multiplies opposite sides: $A B$ with $C D$ , and $A D$ with $B C$ . Two adjacent sides share a vertex, so they are not a pair.

A B \cdot C D + A D \cdot B C = A C \cdot B D

. A quick check: in every product, each of the four letters

A

B

C

D

should appear exactly once.

“Each product uses all four vertices exactly once”

Exhibit

Four points on a circle, but in the wrong order

A

B

C

D

lie on one circle, so

A B \cdot C D + A D \cdot B C = A C \cdot B D

What went wrong? Think first, then open

Diagnosis The theorem is about a cyclic quadrilateral $A B C D$ : the four points lie on the circle in the order $A$ , $B$ , $C$ , $D$ , so $A C$ and $B D$ are the diagonals. If the order around the circle is $A$ , $D$ , $B$ , $C$ , then $A C$ is a side and the equation fails.

First read off the order of the four points around the circle and decide which two segments are the diagonals (they cross inside the circle). Then write the equation with the product of the diagonals on its own side.

“First work out which segments are the diagonals”