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Geometry: Ptolemy's Theorem

Ptolemy's Theorem states that, in a cyclic quadrilateral, the product of the diagonals is equal to the sum the products of the opposite sides.

In the diagram below, Ptolemy's Theorem claims:

Proof

Consider the diagram on the right:

We have drawn a quadrilateral ABCD inside a circle, and constructed the angle ABK so it is equal to angle DBC.

We note the following facts about the angles in the same segment:

  • (The chord AD subtends equal angles in the same segment.) The angles are marked with one stroke in the diagram.
  • (The chord BC subtends equal angles in the same segment.) The angles are marked with two strokes in the diagram.
  • (The chord DC subtends equal angles in the same segment.) The angles are marked with three strokes in the diagram.

And the following about the named triangles:

  • Triangles ABK and DBC are similar, sim1 because two of the angles in each are equal (as marked in the diagram), and the third angles are equal because of the angle sum of a triangle.
  • The triangles ABC and KBC are similar, sim2, because the angles are equal (as marked in the diagram).
ptolemy's Theorem

The diagram is repeated on the right to minimize scrolling.

From the triangles ABK and DBC we have:

[1.01]

From the triangles ABC and KBC, we have:

[1.02]

Adding 1.01 and 1.02 (and taking out the common factor DB) we have:

[1.03]

Because AK and CK are the parts of AC, we have Ptolemy's Theorem:

[1.04]

ptolemy's Theorem

 


Ken Ward's Mathematics Pages



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