Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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4b(301)

Malfatti circles

4b (301)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&, \\ P_{\mathbf{4b}}&{}\approx{}&0.005430340741&{}:{}&-0.003382415975&{}:{}&0.997952075234&, \\ P^-_{\mathbf{4b}}&{}\approx{}&-0.007105785090&{}:{}&0.017630636481&{}:{}&0.989475148609&, \\ P^+_{\mathbf{4b}}&{}\approx{}&0.017459223766&{}:{}&-0.023545228107&{}:{}&1.006086004341&, \\ Q_{\mathbf{4b}}&{}\approx{}&-0.268633447366&{}:{}&0.054741213878&{}:{}&1.213892233488&, \\ I^\prime_{\mathbf{4b}}&{}\approx{}&0.048335533334&{}:{}&-0.049248256203&{}:{}&1.000912722870&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{4b}}\)
\(P^-_{\mathbf{4b}}\)
\(P^+_{\mathbf{4b}}\)
\(Q_{\mathbf{4b}}\)
\(I^\prime_{\mathbf{4b}}\)
4b (301)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4b}}&{}\approx{}&-5.350893551327&{}:{}&-15.877233878318&{}:{}&22.228127429646&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.211551064408&{}:{}&0.294829785307&{}:{}&0.493619150285&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.012895094052&{}:{}&-0.021491823420&{}:{}&1.008596729368&, \\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.024308465701&{}:{}&-0.003318213663&{}:{}&0.979009747963&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.005644349521&{}:{}&-0.042925541745&{}:{}&1.037281192224&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.090709597619&{}:{}&-0.056500615106&{}:{}&0.965791017487&, \\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.012405532399&{}:{}&0.017289066657&{}:{}&0.970305400944&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.007391965055&{}:{}&-0.021933525096&{}:{}&1.029325490151&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.177454008928&{}:{}&0.440292956195&{}:{}&0.737161052733&, \\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.043149748252&{}:{}&0.956850251748&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.284190373377&{}:{}&0.000000000000&{}:{}&1.284190373377&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.255928946018&{}:{}&-0.255928946018&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4b}}}{B^\prime_{\mathbf{4b}}}{C^\prime_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^*_{\mathbf{4b}}}{B^*_{\mathbf{4b}}}{C^*_{\mathbf{4b}}}\)
4b (301)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4b}}}}&{}\approx{}&15.877233878318&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4b}}}}&{}\approx{}&0.352585107347&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4b}}}}&{}\approx{}&0.021491823420&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-5.350893551327&{}:{}&-15.877233878318&{}:{}&22.228127429646&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.211551064408&{}:{}&0.294829785307&{}:{}&0.493619150285&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.012895094052&{}:{}&-0.021491823420&{}:{}&1.008596729368&. \end{alignedat} \]
4b (301)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4b}}&{}\approx{}&0.005430340741&{}:{}&-0.003382415975&{}:{}&0.997952075234&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.024308465701&{}:{}&-0.003318213663&{}:{}&0.979009747963&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.005644349521&{}:{}&-0.042925541745&{}:{}&1.037281192224&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.090709597619&{}:{}&-0.056500615106&{}:{}&0.965791017487&. \end{alignedat} \]
4b (301)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4b}}&{}\approx{}&-0.007105785090&{}:{}&0.017630636481&{}:{}&0.989475148609&,\\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.012405532399&{}:{}&0.017289066657&{}:{}&0.970305400944&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.007391965055&{}:{}&-0.021933525096&{}:{}&1.029325490151&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.177454008928&{}:{}&0.440292956195&{}:{}&0.737161052733&. \end{alignedat} \]
4b (301)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4b}}&{}\approx{}&0.017459223766&{}:{}&-0.023545228107&{}:{}&1.006086004341&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-5.350893551327&{}:{}&-15.877233878318&{}:{}&22.228127429646&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.211551064408&{}:{}&0.294829785307&{}:{}&0.493619150285&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.012895094052&{}:{}&-0.021491823420&{}:{}&1.008596729368&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.024308465701&{}:{}&-0.003318213663&{}:{}&0.979009747963&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.005644349521&{}:{}&-0.042925541745&{}:{}&1.037281192224&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.090709597619&{}:{}&-0.056500615106&{}:{}&0.965791017487&, \end{alignedat} \]
4b (301)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4b}}&{}\approx{}&-0.268633447366&{}:{}&0.054741213878&{}:{}&1.213892233488&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.043149748252&{}:{}&0.956850251748&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.284190373377&{}:{}&0.000000000000&{}:{}&1.284190373377&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.255928946018&{}:{}&-0.255928946018&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4b}}&{}\approx{}&0.048335533334&{}:{}&-0.049248256203&{}:{}&1.000912722870&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.024308465701&{}:{}&-0.003318213663&{}:{}&0.979009747963&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.005644349521&{}:{}&-0.042925541745&{}:{}&1.037281192224&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.090709597619&{}:{}&-0.056500615106&{}:{}&0.965791017487&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.043149748252&{}:{}&0.956850251748&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.284190373377&{}:{}&0.000000000000&{}:{}&1.284190373377&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.255928946018&{}:{}&-0.255928946018&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)