There are three circles C1, C2, C3 with radii r1, r2, r3 (r1 < r2 < r3) touches each other externally as shown and L1, L2 are two common tangents to these circles from point P. Which of the following is(are) correct?

Engineering

Mathematics

Common Tangent To two Circle

Question

There are three circles C₁, C₂, C₃ with radii r₁, r₂, r₃ (r₁ < r₂ < r₃) touches each other externally as shown and L₁, L₂ are two common tangents to these circles from point P.

Which of the following is(are) correct?

r₂² = r₁r₃

$DE = \sqrt{r_{2}^{2} - r_{1}^{2}}$

$DE = 2 \sqrt{r_{1} r_{2}}$

$r_{2} = \frac{r_{1}^{2} + r_{3}^{2}}{r_{1} + r_{3}}$

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$DE = AK = \sqrt{{(r_{1} + r_{2})}^{2} - {(r_{1} - r_{2})}^{2}}$

$= 2 \sqrt{r_{1} r_{2}}$

Let PD = x and $∠$ APD = θ

$∴ tanθ = \frac{AD}{PD} = \frac{BE}{PE} = \frac{CF}{PF}$

$\Rightarrow \frac{r_{1}}{x} = \frac{r_{2}}{x + 2 \sqrt{r_{1} r_{2}}}$

$= \frac{r_{3}}{x + 2 \sqrt{r_{1} r_{2}} + 2 \sqrt{r_{2} r_{3}}}$

$\Rightarrow \frac{r_{2} - r_{1}}{2 \sqrt{r_{1} r_{2}}} = \frac{r_{3} - r_{2}}{2 \sqrt{r_{2} r_{3}}}$

$\Rightarrow r_{2} = \sqrt{r_{1} r_{3}} .$