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 caΟβbβ, 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+242β=202+152β=25
If AC<25, then β ABC and β ADC are both acute, so ABCD cannot be cyclic. Analogously, if AC>25, then β ABC and β ADC are both obtuse, and again ABCD cannot be cyclic. Therefore β³ABC and β³CDA are both right triangles with hypotenuse 25 .
The area of ABCD is 21β(7β 24+15β 20)=234. Because β ABC and β ADC are right angles, AC is the diameter of the circumcircle, so the circumcircle has radius 225β and its area is
