Ptolemy Theorem

Ptolemy Theorem states that in an inscribed quadrilateral, the sum of the products of its opposite sides is equal to the product of its diagonals, or ac + bd = ef

(1) For each side(a, b, c, and d) and its corresponding inscribed angle (A, B, C, and D) of the inscribed qudrilateral, we have :

a = 2R sin A
b = 2R sin B
c = 2R sin C

For each diagonal (e and f) and its corresponding inscribed angle (E and F) of the inscribed qudrilateral, we have:

e = 2R sin E, E = A + B
f = 2R sin F, F = A + D

(2) In order to prove ac + bd = ef, we have to prove:

sin A sin C + sin B sin D = sin (A+B) sin (A+D)

sin A sin C = 0.5 [ cos (A-C) - cos (A+C) ]
sin B sin D = 0.5 [ cos (B-D) - cos (B+D) ]

(3) If we notice that

(A+C) + (B+D) = 180
cos (A+C) = - cos (B+D)

then we have:

sin A sin C + sin B sin D = 0.5 [ cos (A-C) + cos (B-D) ]

(4) If we also notice that

A + B + A + D=A +(180-C) = 180 + (A-C)
cos (180+A-C) = -cos(A-C)

sin (A+B) sin (A+D) = 0.5 [ cos (B-D) + cos (A-C) ]

(5) So, sin A sin C + sin B sin D = sin (A+B) sin (A+D)

(6) Therefore, ad + cd = ef

CHRISRAO/饶旭东