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
Best Answer
"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.