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,
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, ,
because the angles are equal (as marked in the diagram).
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:
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