Ptolemy’s Theorem
by aoum, Apr 16, 2025, 1:06 AM
Ptolemy’s Theorem: A Classic Result in Euclidean Geometry
Ptolemy’s Theorem is a foundational result in cyclic geometry that describes a precise relationship between the sides and diagonals of a cyclic quadrilateral (a quadrilateral inscribed in a circle). Named after the ancient Greek mathematician Claudius Ptolemy, it lies at the heart of trigonometry and classical geometry.
![[asy]
size(200);
pair A = dir(110);
pair B = dir(40);
pair C = dir(-20);
pair D = dir(210);
draw(circle((0,0),1));
draw(A--B--C--D--cycle);
draw(A--C, dashed+blue);
draw(B--D, dashed+blue);
label("$A$", A, NW);
label("$B$", B, NE);
label("$C$", C, SE);
label("$D$", D, SW);
label("$AC$", midpoint(A--C), NE);
label("$BD$", midpoint(B--D), SW);
label("$AB$", midpoint(A--B), N);
label("$BC$", midpoint(B--C), E);
label("$CD$", midpoint(C--D), S);
label("$DA$", midpoint(D--A), W);
[/asy]](//latex.artofproblemsolving.com/a/c/8/ac85fa424af56f933842e2a13e669fe660964760.png)
1. Statement of Ptolemy’s Theorem
Let
be a cyclic quadrilateral. Then the product of the two diagonals equals the sum of the products of the two pairs of opposite sides:
This relation is exact if and only if the quadrilateral is inscribed in a circle.
2. Applications of Ptolemy’s Theorem
3. Special Cases
If is a rectangle (so 
and 
), then the theorem reduces to the Pythagorean Theorem:
This is because in a rectangle inscribed in a circle, the diagonals are equal and intersect at right angles.
4. Proof Using Similar Triangles
Let’s outline a synthetic geometric proof:
Construct point
on 
such that 
. Then triangles 
and 
are similar (angle-angle). Therefore:
Similarly, one shows:
Now since lies on , and 
, adding gives:
Multiplying both sides by
and simplifying yields the result:
5. Ptolemy's Inequality
If the quadrilateral is not cyclic, the relation becomes an inequality:
with equality if and only if the quadrilateral is cyclic.
6. Coordinate Proof (Analytic Geometry)
Place the cyclic quadrilateral on the unit circle using complex numbers:
Let
, 
, 
, and 
with 
. Then the chord lengths correspond to:
Ptolemy’s Theorem follows from identities involving complex absolute values and angles.
7. Trigonometric Form of Ptolemy’s Theorem
If is inscribed in a circle, and the arcs subtended by sides are known, then the theorem can also be written using the law of sines:
8. Connections and Historical Note
Ptolemy used this theorem extensively in his work Almagest to create a table of chords — the precursor to modern trigonometric functions. In that setting, he used it to find exact values for chords of angles such as
and 
.
9. References
Ptolemy’s Theorem is a foundational result in cyclic geometry that describes a precise relationship between the sides and diagonals of a cyclic quadrilateral (a quadrilateral inscribed in a circle). Named after the ancient Greek mathematician Claudius Ptolemy, it lies at the heart of trigonometry and classical geometry.
![[asy]
size(200);
pair A = dir(110);
pair B = dir(40);
pair C = dir(-20);
pair D = dir(210);
draw(circle((0,0),1));
draw(A--B--C--D--cycle);
draw(A--C, dashed+blue);
draw(B--D, dashed+blue);
label("$A$", A, NW);
label("$B$", B, NE);
label("$C$", C, SE);
label("$D$", D, SW);
label("$AC$", midpoint(A--C), NE);
label("$BD$", midpoint(B--D), SW);
label("$AB$", midpoint(A--B), N);
label("$BC$", midpoint(B--C), E);
label("$CD$", midpoint(C--D), S);
label("$DA$", midpoint(D--A), W);
[/asy]](http://latex.artofproblemsolving.com/a/c/8/ac85fa424af56f933842e2a13e669fe660964760.png)
Ptolemy's theorem is a relation among these lengths in a cyclic quadrilateral.
1. Statement of Ptolemy’s Theorem
Let
be a cyclic quadrilateral. Then the product of the two diagonals equals the sum of the products of the two pairs of opposite sides:
This relation is exact if and only if the quadrilateral is inscribed in a circle.
2. Applications of Ptolemy’s Theorem
- Deriving trigonometric identities
- Proving special cases of triangle relationships
- Solving geometry olympiad problems
- Constructing chord lengths in circle geometry
3. Special Cases
If
is a rectangle (so 
and 
), then the theorem reduces to the Pythagorean Theorem:
This is because in a rectangle inscribed in a circle, the diagonals are equal and intersect at right angles.
4. Proof Using Similar Triangles
Let’s outline a synthetic geometric proof:
Construct point
on 
such that 
. Then triangles 
and 
are similar (angle-angle). Therefore:
Similarly, one shows:
Now since
lies on , and 
, adding gives:
Multiplying both sides by
and simplifying yields the result:5. Ptolemy's Inequality
If the quadrilateral
is not cyclic, the relation becomes an inequality:
with equality if and only if the quadrilateral is cyclic.
6. Coordinate Proof (Analytic Geometry)
Place the cyclic quadrilateral on the unit circle using complex numbers:
Let
, 
, 
, and 
with 
. Then the chord lengths correspond to:
Ptolemy’s Theorem follows from identities involving complex absolute values and angles.
7. Trigonometric Form of Ptolemy’s Theorem
If
is inscribed in a circle, and the arcs subtended by sides are known, then the theorem can also be written using the law of sines:
8. Connections and Historical Note
Ptolemy used this theorem extensively in his work Almagest to create a table of chords — the precursor to modern trigonometric functions. In that setting, he used it to find exact values for chords of angles such as
and 
.9. References
- Wikipedia: Ptolemy’s Theorem
- AoPS Wiki: Ptolemy’s Theorem
- Euclid's Elements, Book XIII
- Courant and Robbins, What is Mathematics?
April 2025
March 2025