How to prove $\frac{1}{x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+2\sqrt{\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}}$

You can prove this equation using geometry approach. Note that this equation is exactly similar with Descartes 4 Circles Theorem. Descartes' theorem says: If four circles are tangent to each other at six distinct points and the circles have curvatures $k_i$ (for $i = 1,\cdots, 4$), then $k_i$ satisfies the following relation: $$ (k_1+k_2+k_3+k_4)^2=2(k_1^2+k_2^2+k_3^2+k_4^2), $$ where $k_i=\pm\dfrac{1}{r_i}$, $r_i$ is the radius of circle. The equation can also be written as: $$ k_4=k_1+k_2+k_3\pm2\sqrt{k_1k_2+k_2k_3+k_1k_3}, $$ or $$ \frac{1}{r_4}=\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}\pm2\sqrt{\frac{1}{r_1r_2}+\frac{1}{r_2r_3}+\frac{1}{r_1r_3}}. $$ The generalization to $n$ dimensions or variables is referred to as the Soddy–Gosset theorem. $$ \left(\sum_{i=1}^{n+2}k_i\right)^2=n\sum_{i=1}^{n+2}k_i^2. $$ For detail explanation and complete proof of Descartes' theorem (also to answer your question), you may refer to these sites: 1, 2, or download this journal. $$\\$$

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