Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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2b(121)

Malfatti circles

2b (121)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&, \\ P_{\mathbf{2b}}&{}\approx{}&-0.003640020030&{}:{}&1.314880716372&{}:{}&-0.311240696342&, \\ P^-_{\mathbf{2b}}&{}\approx{}&0.147266920696&{}:{}&0.736172288398&{}:{}&0.116560790906&, \\ P^+_{\mathbf{2b}}&{}\approx{}&-0.305447773169&{}:{}&2.472274071036&{}:{}&-1.166826297867&, \\ Q_{\mathbf{2b}}&{}\approx{}&-1.212072001599&{}:{}&5.948042748385&{}:{}&-3.735970746785&, \\ I^\prime_{\mathbf{2b}}&{}\approx{}&-0.106249659921&{}:{}&2.607013108030&{}:{}&-1.500763448109&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{2b}}\)
\(P^-_{\mathbf{2b}}\)
\(P^+_{\mathbf{2b}}\)
\(Q_{\mathbf{2b}}\)
\(I^\prime_{\mathbf{2b}}\)
2b (121)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2b}}&{}\approx{}&3.115510644079&{}:{}&5.288776610198&{}:{}&-7.404287254278&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.635089355133&{}:{}&3.116964517109&{}:{}&-1.481875161976&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.692638588899&{}:{}&1.154397648165&{}:{}&0.538240940734&, \\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.662616760822&{}:{}&2.178213975023&{}:{}&-0.515597214200&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.010347789033&{}:{}&1.895137718045&{}:{}&-0.884789929012&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.009631399157&{}:{}&3.479140476367&{}:{}&-2.469509077210&, \\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.213447051941&{}:{}&1.047579969346&{}:{}&0.165867082595&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.286610418828&{}:{}&0.486539335764&{}:{}&0.226850245409&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.276066993599&{}:{}&1.380030691676&{}:{}&-0.656097685275&, \\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.688901059317&{}:{}&-1.688901059317&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.244959888836&{}:{}&0.000000000000&{}:{}&0.755040111164&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.255928946018&{}:{}&1.255928946018&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2b}}}{B^\prime_{\mathbf{2b}}}{C^\prime_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^*_{\mathbf{2b}}}{B^*_{\mathbf{2b}}}{C^*_{\mathbf{2b}}}\)
2b (121)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2b}}}}&{}\approx{}&-5.288776610198&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.058482258555&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.154397648165&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&3.115510644079&{}:{}&5.288776610198&{}:{}&-7.404287254278&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.635089355133&{}:{}&3.116964517109&{}:{}&-1.481875161976&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.692638588899&{}:{}&1.154397648165&{}:{}&0.538240940734&. \end{alignedat} \]
2b (121)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2b}}&{}\approx{}&-0.003640020030&{}:{}&1.314880716372&{}:{}&-0.311240696342&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.662616760822&{}:{}&2.178213975023&{}:{}&-0.515597214200&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.010347789033&{}:{}&1.895137718045&{}:{}&-0.884789929012&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.009631399157&{}:{}&3.479140476367&{}:{}&-2.469509077210&. \end{alignedat} \]
2b (121)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2b}}&{}\approx{}&0.147266920696&{}:{}&0.736172288398&{}:{}&0.116560790906&,\\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.213447051941&{}:{}&1.047579969346&{}:{}&0.165867082595&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.286610418828&{}:{}&0.486539335764&{}:{}&0.226850245409&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.276066993599&{}:{}&1.380030691676&{}:{}&-0.656097685275&. \end{alignedat} \]
2b (121)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2b}}&{}\approx{}&-0.305447773169&{}:{}&2.472274071036&{}:{}&-1.166826297867&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&3.115510644079&{}:{}&5.288776610198&{}:{}&-7.404287254278&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.635089355133&{}:{}&3.116964517109&{}:{}&-1.481875161976&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.692638588899&{}:{}&1.154397648165&{}:{}&0.538240940734&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.662616760822&{}:{}&2.178213975023&{}:{}&-0.515597214200&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.010347789033&{}:{}&1.895137718045&{}:{}&-0.884789929012&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.009631399157&{}:{}&3.479140476367&{}:{}&-2.469509077210&, \end{alignedat} \]
2b (121)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2b}}&{}\approx{}&-1.212072001599&{}:{}&5.948042748385&{}:{}&-3.735970746785&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.688901059317&{}:{}&-1.688901059317&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.244959888836&{}:{}&0.000000000000&{}:{}&0.755040111164&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.255928946018&{}:{}&1.255928946018&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2b}}&{}\approx{}&-0.106249659921&{}:{}&2.607013108030&{}:{}&-1.500763448109&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.662616760822&{}:{}&2.178213975023&{}:{}&-0.515597214200&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.010347789033&{}:{}&1.895137718045&{}:{}&-0.884789929012&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.009631399157&{}:{}&3.479140476367&{}:{}&-2.469509077210&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.688901059317&{}:{}&-1.688901059317&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.244959888836&{}:{}&0.000000000000&{}:{}&0.755040111164&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.255928946018&{}:{}&1.255928946018&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)