Question

Two circle with radii r₁ and r₂ touch each other externally. Let r be the radius of a circle which touches these two circle as well as a common tangent to the two circles, Prove that: `1/sqrtr + 1/sqrtr_1 + 1/sqrtr_2`.

Answer 1

From the adjoining figure,
PQ = SY = `sqrt( "XY"^2 - "XS"^2)`
= `sqrt((r_1 + r_2)^2 - (r_1 - r_2)^2)`
= `sqrt(4r_1r_2)`
= `sqrt(r_1r_2)`

Similarly, PR = `2 sqrt(rr_1)` and RQ = `2 sqrt(rr_2)`
Now, PQ = PR + RQ
`2 sqrt(r_1r_2) = 2 sqrt(rr_1) = 2 sqrt(rr_2) `

⇒ `sqrt(r_1r_2) = sqrt(rr_1) = sqrt(rr_2) `

Dividing by `sqrt(rr_1r_2)` on both sides,

⇒ `1/sqrtr + 1/sqrtr_1 + 1/sqrtr_2`.
Hence proved.