Problem:
Quadrilateral ABCD with side lengths AB=7,BC=24,CD=20,DA=15 is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form caπ−b​, where a,b, and c are positive integers such that a and c have no common prime factor. What is a+b+c?
Answer Choices:
A. 260
B. 855
C. 1235
D. 1565
E. 1997
Solution:
Observe that
72+242​=202+152​=25
If AC<25, then ∠ABC and ∠ADC are both acute, so ABCD cannot be cyclic. Analogously, if AC>25, then ∠ABC and ∠ADC are both obtuse, and again ABCD cannot be cyclic. Therefore △ABC and △CDA are both right triangles with hypotenuse 25 .
The area of ABCD is 21​(7⋅24+15⋅20)=234. Because ∠ABC and ∠ADC are right angles, AC is the diameter of the circumcircle, so the circumcircle has radius 225​ and its area is
