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# Three circles touch the one another externally. The tangents at their point of contact meet at a point whose distance from a point of contanct is 4. Find the product of the radii to the sum of the radii of the circles

Jitender Pal
6 years ago
Hello Student,
Let us consider circles with centers at A, B and C and with radii r1, r2and r3respectively which touch each other externally at P, Q and R. Let the common tangents at P, Q and R meet each other at O. Then OP = OQ = OR = 4 (given)(lengths of tangents from a pt to a circle are equal).
Also OP ⊥ AB, OQ ⊥ AC, OR ⊥BC.
⇒ O is the in centre of the ∆ABC Thus for ∆ ABC
s = (r1 + r2) + (r2 + r3) + (r3 + r1)/2
i.e. s = (r1 + r2 + r3)
∴ ∆ = √(r1 + r2 + r3) r1 r2 r3 (Heron’s formula)
Now r = ∆/s
NOTE THIS STEP :
⇒ 4 = √(r1 + r2 + r3) r1 r2 r3/r1 + r2 + r3
⇒ 4 = √r1 r2 r3/√r1 + r2 + r3
⇒ r1 r2 r3/r1 + r2 r3 = 16/1
⇒ r1. r2. r3 : r1 + r2 + r3 = 16 : 1

Thanks
Jitender Pal