Problem:
Circles Ο1β and Ο2β intersect at two points P and Q, and their common tangent line closer to P intersects Ο1β and Ο2β at points A and B, respectively. The line parallel to AB that passes through P intersects Ο1β and Ο2β for the second time at points X and Y, respectively. Suppose PX=10,PY=14, and PQ=5. Then the area of trapezoid XABY is mnβ, where m and n are positive integers and n is not divisible by the square of any prime. Find m+n.
Solution:
Answer (033):
Let M be the intersection of line PQ with AB, and let D,E, and F be the projections of A,M, and B, respectively, onto XY. Because ABβ₯XY, points A and B are the midpoints of \wideparen{X P} and \wideparen{Y P}, respectively, so D is the midpoint of XP and F is the midpoint of YP. This implies AB=DF=21ββ XY=12. The power of the point M with respect to Ο1β and with respect to Ο2β are both MPβ MQ implying that AM2=MPβ MQ=MB2, so AM=MB=21ββ AB=6. Thus
36=AM2=MPβ MQ=MP(MP+5),
which implies that MP=4.
Finally, observe that DE=AM=6, so PE=DEβDP=6β21ββ 10=1 and ME=PM2βPE2β=15β. It follows that the area of trapezoid XABY is