inscribed circles inside circle
Mia Lopez 
Found answer to 3 inscribed tangent circles inside another circle, but solution used trig. This question is about 2 inscribed tangent circles ON THE DIAMETER OF THE LARGEST CIRCLE, and this GRE question cannot use trig as a solution. The answer seems intuitive, but I can't explain it.
"Three circles with centers on line segment PQ are tangent at points P, Q, R, where point R lies on line segment PQ." (PQ is diameter of largest circle.)
Which is greater (or equal)? Quantity A: Circumference of largest circle? Quantity B: Sum of circumference of 2 smaller inscribed circles?
Set D=10 for larger circle. Makes r=5 (becomes diameter of smaller circles). Qty A: $C=\pi D$ $C=10 \pi = 31.42$ (Circumference of larg circle) Qty B: Sum of $\pi (d)$ (small diameter) = 2[(Pd)(d)] = 31.42 Answer is: QtyA = QtyB
- IS THERE A RULE IN GEOMETRY about tangent circles on the Diameter of circle? I haven't found one.
- Am I missing something? Thanks. Charlie
2 Answers
$\begingroup$"Inscribed" is the key word. You necessarily have despite $d_1$ and $d_2$ can vary a lot $$D=d_1+d_2$$ where $D,d_1,d_2$ are the diameters. Thus the equivalent equality $$D\pi=d_1\pi+d_2\pi$$ Consequently the sums are equal.
$\endgroup$ $\begingroup$The main thing here is not that the two smaller circles are on the diameter of the larger circle and tangent with each other and the larger circle. That's just to let you know you have two circles, the sum of whose diameters equals the diameter of a larger circle. Since you're asked to compare the circumference of the larger with the sum of the circumferences of the two smaller circles, and since a circle's circumference is proportional to its diameter, you have$$\frac{C}{c_1}=\frac{D}{d_1}$$from which, by separation of ratios $$\frac{C}{C-c_1}=\frac{D}{D-d_1}$$ But $$D-d_1=d_2$$ Therefore $$C-c_1=c_2$$ Hence the circumference of the larger circle equals the sum of the circumferences of the two smaller circles
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