Three circles of radii a,b,c (a

Question

Three circles of radii $a,b,c$ ($a<b<c$) touch each other externally. If they have the $x$-axis as a common tangent, then:

Accepted Answer

AB = AC + CB

$$\sqrt {{{\left( {b + c} \right)}^2} - {{\left( {c - b} \right)}^2}} $$ = $$\sqrt {{{\left( {b + a} \right)}^2} - {{\left( {b - a} \right)}^2}} $$

+ $$\sqrt {{{\left( {c + a} \right)}^2} - {{\left( {c - a} \right)}^2}} $$

$$ \Rightarrow $$  $$\sqrt {2bc} $$ = $$\sqrt {2ac} $$ + $$\sqrt {2ab} $$

Dividing by $$\sqrt {abc} $$ we get.

$$ \Rightarrow $$  $${1 \over {\sqrt a }}$$ = $${1 \over {\sqrt b }}$$ + $${1 \over {\sqrt c }}$$

Solutions · 1