Discussion: Descartes Four Circle Theorem

I noticed that many people have been posting problems involving Descartes' Four Circle Theorem, but this site has no documentation on it (especially the proof of it). It seems that all the proofs presented online are by Coxeter, which is short but non-elementary. I will attempt to prove the theorem from scratch and see where I get in the next few days. For the time being, here is Coxeter's proof.

Interesting Trivia

Descartes found the theorem without proof and sent it to Princess Elizabeth of Bohemia in 1643. The theorem was first proven by Jakob Steiner nearly 200 years later in 1826. Interestingly, this theorem was known to the Japanese around 1830 in Hashimoto Masataka's book Sanpo Tenzan Shogakusho (筭法點竄初學抄).

Key Equations

${r}_{4}=\frac{{r}_{1}{r}_{2}{r}_{3}}{{r}_{1}{r}_{2}+{r}_{1}{r}_{3}+{r}_{2}{r}_{3}+2\sqrt{{r}_{1}{r}_{2}{r}_{3}({r}_{1}+{r}_{2}+{r}_{3})}}$

${R}_{4}=\frac{-{r}_{1}{r}_{2}{r}_{3}}{{r}_{1}{r}_{2}+{r}_{1}{r}_{3}+{r}_{2}{r}_{3}-2\sqrt{{r}_{1}{r}_{2}{r}_{3}({r}_{1}+{r}_{2}+{r}_{3})}}$

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Discussion: Descartes Four Circle Theorem

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