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

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