Once upon a time, Ptolemy drew 2 red perpendicular lines of length and and said, "This is a special intersection as the lengths from point to other vertices are all in integers."
Brahmagupta then drew 4 blue lines, connecting the vertices, and created a quadrilateral. "This is a unique quadrilateral," he claimed, "as the 4 side lengths are all in integers."
Finally, Parameshvara drew a circle, circumscribing that quadrilateral before declaring, "This is a wonderful circle, for its diameter also has the length in integer."
What is the length of this circumcircle's diameter according to these 3 wisemen?
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Let A E = a and B E = b for some integers a , b . Then C E = 5 6 − a and D E = 6 3 − b .
According to the chord theorem, A E ⋅ C E = B E ⋅ D E .
Thus, a ( 5 6 − a ) = b ( 6 3 − b ) __(1)
Then by Ptolemy's Theorem , A C ⋅ B D = A B ⋅ C D + A D ⋅ B C
By Pythagorean Theorem, A B = a 2 + b 2 , B C = ( 5 6 − a ) 2 + b 2 , C D = ( 5 6 − a ) 2 + ( 6 3 − b ) 2 , D A = a 2 + ( 6 3 − b ) 2 .
Thus, 5 6 ⋅ 6 3 = ( a 2 + b 2 ) [ ( 5 6 − a ) 2 + ( 6 3 − b ) 2 ] + [ ( 5 6 − a ) 2 + b 2 ] [ a 2 + ( 6 3 − b ) 2 ] _(2)
Solving for the 2 equations, we will obtain the side lengths of quadrilateral: 2 5 , 3 9 , 6 0 , 5 2 .
The area of the quadrilateral is 2 1 ( 5 6 × 6 3 = 1764)
Then by using Parameshvara's formula, we can compute the radius of the circumcircle as:
R = 4 × 1 7 6 4 1 ( 2 5 × 3 9 + 6 0 × 5 2 ) + ( 2 5 × 6 0 + 3 9 × 5 2 ) + ( 2 5 × 5 2 + 3 9 × 6 0 ) = 2 6 5 .
Therefore, the diameter of the circumcircle equals 6 5 .