Circle C with radius 2 has diameter AB¯. Circle D is internally tangent to circle C at A. Circle E is internally tangent to circle C, externally tangent to circle D, and tangent to AB¯. The radius of circle D is three times the radius of circle E, and can be written in the form m−n, where m and n are positive integers. The value of m + n _____

DA = 3r, EF = r, CE = 2, CD = 2 – 3r, DE = 3r + r = 4rsin∠EDC=EFDE=14⇒cos∠EDC=154Cosine law of ΔDEC, (24+615)r2=(8+415)r⇒3r=415−14⇒254.

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Circle C with radius 2 has diameter $\bar{A B} .$ Circle D is internally tangent to circle C at A. Circle E is internally tangent to circle C, externally tangent to circle D, and tangent to $\bar{A B} .$ The radius of circle D is three times the radius of circle E, and can be written in the form $\sqrt{m} - n,$ where m and n are positive integers. The value of m + n _____

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answer is 254.

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Detailed Solution

DA = 3r, EF = r, CE = 2, CD = 2 – 3r, DE = 3r + r = 4r

$\sin ∠ E D C = \frac{E F}{D E} = \frac{1}{4} \Rightarrow \cos ∠ E D C = \frac{\sqrt{15}}{4}$

Cosine law of $Δ D E C, (24 + 6 \sqrt{15}) r^{2} = (8 + 4 \sqrt{15}) r \Rightarrow 3 r = 4 \sqrt{15} - 14 \Rightarrow 254 .$

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