Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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7b(323)

Malfatti circles

7b (323)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&, \\ P_{\mathbf{7b}}&{}\approx{}&-0.001291273044&{}:{}&1.002514509734&{}:{}&-0.001223236690&, \\ P^-_{\mathbf{7b}}&{}\approx{}&0.006100425007&{}:{}&0.977897517502&{}:{}&0.016002057491&, \\ P^+_{\mathbf{7b}}&{}\approx{}&-0.008869284705&{}:{}&1.027751992775&{}:{}&-0.018882708070&, \\ Q_{\mathbf{7b}}&{}\approx{}&0.117387866014&{}:{}&0.844527642600&{}:{}&0.038084491386&, \\ I^\prime_{\mathbf{7b}}&{}\approx{}&-0.029864174568&{}:{}&1.074264393877&{}:{}&-0.044400219308&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{7b}}\)
\(P^-_{\mathbf{7b}}\)
\(P^+_{\mathbf{7b}}\)
\(Q_{\mathbf{7b}}\)
\(I^\prime_{\mathbf{7b}}\)
7b (323)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7b}}&{}\approx{}&5.155831533393&{}:{}&10.389578833482&{}:{}&-14.545410366875&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.007737056431&{}:{}&1.025790188103&{}:{}&-0.018053131672&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.352585107347&{}:{}&0.587641845578&{}:{}&0.764943261769&, \\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.015141843309&{}:{}&1.016382000675&{}:{}&-0.001240157367&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.057703896919&{}:{}&1.112367412296&{}:{}&-0.054663515377&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.001336203800&{}:{}&1.037397709170&{}:{}&-0.036061505370&, \\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.007476580644&{}:{}&0.991255929614&{}:{}&0.016220651030&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.177361018291&{}:{}&0.357402345205&{}:{}&0.465236636504&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.006309994529&{}:{}&1.011491490884&{}:{}&-0.017801485413&, \\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.956850251748&{}:{}&0.043149748252&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.755040111164&{}:{}&0.000000000000&{}:{}&0.244959888836&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.122035526991&{}:{}&0.877964473009&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7b}}}{B^\prime_{\mathbf{7b}}}{C^\prime_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^*_{\mathbf{7b}}}{B^*_{\mathbf{7b}}}{C^*_{\mathbf{7b}}}\)
7b (323)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7b}}}}&{}\approx{}&-10.389578833482&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.012895094052&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.587641845578&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&5.155831533393&{}:{}&10.389578833482&{}:{}&-14.545410366875&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.007737056431&{}:{}&1.025790188103&{}:{}&-0.018053131672&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.352585107347&{}:{}&0.587641845578&{}:{}&0.764943261769&. \end{alignedat} \]
7b (323)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7b}}&{}\approx{}&-0.001291273044&{}:{}&1.002514509734&{}:{}&-0.001223236690&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.015141843309&{}:{}&1.016382000675&{}:{}&-0.001240157367&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.057703896919&{}:{}&1.112367412296&{}:{}&-0.054663515377&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.001336203800&{}:{}&1.037397709170&{}:{}&-0.036061505370&. \end{alignedat} \]
7b (323)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7b}}&{}\approx{}&0.006100425007&{}:{}&0.977897517502&{}:{}&0.016002057491&,\\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.007476580644&{}:{}&0.991255929614&{}:{}&0.016220651030&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.177361018291&{}:{}&0.357402345205&{}:{}&0.465236636504&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.006309994529&{}:{}&1.011491490884&{}:{}&-0.017801485413&. \end{alignedat} \]
7b (323)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7b}}&{}\approx{}&-0.008869284705&{}:{}&1.027751992775&{}:{}&-0.018882708070&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&5.155831533393&{}:{}&10.389578833482&{}:{}&-14.545410366875&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.007737056431&{}:{}&1.025790188103&{}:{}&-0.018053131672&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.352585107347&{}:{}&0.587641845578&{}:{}&0.764943261769&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.015141843309&{}:{}&1.016382000675&{}:{}&-0.001240157367&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.057703896919&{}:{}&1.112367412296&{}:{}&-0.054663515377&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.001336203800&{}:{}&1.037397709170&{}:{}&-0.036061505370&, \end{alignedat} \]
7b (323)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7b}}&{}\approx{}&0.117387866014&{}:{}&0.844527642600&{}:{}&0.038084491386&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.956850251748&{}:{}&0.043149748252&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.755040111164&{}:{}&0.000000000000&{}:{}&0.244959888836&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.122035526991&{}:{}&0.877964473009&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7b}}&{}\approx{}&-0.029864174568&{}:{}&1.074264393877&{}:{}&-0.044400219308&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.015141843309&{}:{}&1.016382000675&{}:{}&-0.001240157367&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.057703896919&{}:{}&1.112367412296&{}:{}&-0.054663515377&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.001336203800&{}:{}&1.037397709170&{}:{}&-0.036061505370&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.956850251748&{}:{}&0.043149748252&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.755040111164&{}:{}&0.000000000000&{}:{}&0.244959888836&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.122035526991&{}:{}&0.877964473009&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)