Ptolemy's Theorem
Let a quadrilateral ABCD be inscribed in a
circle. Then prove the sum of the products of the two pairs of
opposite sides equals the product of its two diagonals. In other
words,
Hint: Locate point M on
BD such that angle ACB = MCD.
Another Suggestion: Ptolemy's Theorem can also be proved from relations known abou the Simson Line. The Simson line results from a Pedal point P located on the circumference of a given triangle ABC. Add line segments BP and CP to the drawing and you have a quadrilateral ABCP with diagonals AC and BP. (See H.S.M. Coxeter and S.L. Greitzer, Geometry Revisited, New Mathematics Library, 1967.
Given a quadrilateral ABCD with the sum of the products of the two pairs of opposited sides equal to the product of the two diagonals. That is, AD·BC + AB·CD = AC·BD. Prove that ABCD is inscribed in a circle.
Let ABC be an equilateral triangle inscribed in a circle. If P is a point on the circumcircle, the of PA, PB, and PC, the larger is always the sum of the two shorter.
Use the Theorem of Ptolemy to prove the Pythagorean theorem.
Use the Theorem of Ptolemy and the Law of Sines to develop the additions and subtractions formulas for sines of angles.