Ptolemy's theorem

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.

If the quadrilateral is given with its four vertices A, B, C, and D in order, then the theorem states that:

where the vertical lines denote the lengths of the line segments between the named vertices. In the context of geometry, the above equality is often simply written as

This relation may be verbally expressed as follows:

Moreover, the converse of Ptolemy's theorem is also true:

Ptolemy's Theorem yields as a corollary a pretty theorem regarding an equilateral triangle inscribed in a circle.

Given An equilateral triangle inscribed on a circle and a point on the circle.

The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices.

Proof: Follows immediately from Ptolemy's theorem:

Any square can be inscribed in a circle whose center is the center of the square. If the common length of its four sides is equal to ${\displaystyle a}$ then the length of the diagonal is equal to ${\displaystyle a{\sqrt {2}}}$ according to the Pythagorean theorem and the relation obviously holds.

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