smc

Question

Circles with centers P, Q and R, having radii 1, 2 and 3, respectively, lie on the same side of line l and are tangent to l at P', Q' and R', respectively, with Q' between P' and R'. The circle with center Q is externally tangent to each of the other two circles. What is the area of triangle PQR?

Accepted Answer

FIGURE Notice that we can find [P'PQRR'] in two different ways: [P'PQQ']+[Q'QRR'] and [PQR]+[P'PRR']. Since we want [PQR], we use the latter method, so we have [P'PQQ']+[Q'QRR']=[PQR]+[P'PRR']. P'Q'=PQ^2-(QQ'-PP')^2=3^2-1^2=8=22. Additionally, Q'R'=QR^2-(RR'-QQ')^2=5^2-1^2=24=26. Therefore, [P'PQQ']=P'P+Q'Q2*22=1+22*22=32. Similarly, [Q'QRR']=56. We can calculate [P'PRR'] easily because P'R'=P'Q'+Q'R'=22+26. [P'PRR']=42+46. Plugging into first equation, the two sums of areas, 32+56=42+46+[PQR], so therefore [PQR]=6-2