Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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5b(303)

Malfatti circles

5b (303)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&, \\ P_{\mathbf{5b}}&{}\approx{}&0.011598854870&{}:{}&-0.003361437575&{}:{}&0.991762582705&, \\ P^-_{\mathbf{5b}}&{}\approx{}&-0.006651931837&{}:{}&0.027551890820&{}:{}&0.979100041017&, \\ P^+_{\mathbf{5b}}&{}\approx{}&0.028783583756&{}:{}&-0.032469068711&{}:{}&1.003685484955&, \\ Q_{\mathbf{5b}}&{}\approx{}&0.236893889655&{}:{}&0.032927836551&{}:{}&0.730178273794&, \\ I^\prime_{\mathbf{5b}}&{}\approx{}&0.069300614852&{}:{}&-0.048163321605&{}:{}&0.978862706753&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{5b}}\)
\(P^-_{\mathbf{5b}}\)
\(P^+_{\mathbf{5b}}\)
\(Q_{\mathbf{5b}}\)
\(I^\prime_{\mathbf{5b}}\)
5b (303)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5b}}&{}\approx{}&-1.155831533393&{}:{}&-5.389578833482&{}:{}&7.545410366875&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.207737056431&{}:{}&0.307543145230&{}:{}&0.484719798339&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.019251774013&{}:{}&-0.032086290022&{}:{}&1.012834516009&, \\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.035237917714&{}:{}&-0.003281043866&{}:{}&0.968043126152&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.012297441176&{}:{}&-0.063792796349&{}:{}&1.051495355173&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.125428466373&{}:{}&-0.036350136674&{}:{}&0.910921670301&, \\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.018151990823&{}:{}&0.026873011709&{}:{}&0.954974997469&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.007065770353&{}:{}&-0.032947298320&{}:{}&1.040013068674&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.103411961942&{}:{}&0.428325959222&{}:{}&0.675086002720&, \\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.043149748252&{}:{}&0.956850251748&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.244959888836&{}:{}&0.000000000000&{}:{}&0.755040111164&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.877964473009&{}:{}&0.122035526991&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5b}}}{B^\prime_{\mathbf{5b}}}{C^\prime_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^*_{\mathbf{5b}}}{B^*_{\mathbf{5b}}}{C^*_{\mathbf{5b}}}\)
5b (303)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5b}}}}&{}\approx{}&5.389578833482&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5b}}}}&{}\approx{}&0.346228427385&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5b}}}}&{}\approx{}&0.032086290022&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&-1.155831533393&{}:{}&-5.389578833482&{}:{}&7.545410366875&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.207737056431&{}:{}&0.307543145230&{}:{}&0.484719798339&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.019251774013&{}:{}&-0.032086290022&{}:{}&1.012834516009&. \end{alignedat} \]
5b (303)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5b}}&{}\approx{}&0.011598854870&{}:{}&-0.003361437575&{}:{}&0.991762582705&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.035237917714&{}:{}&-0.003281043866&{}:{}&0.968043126152&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.012297441176&{}:{}&-0.063792796349&{}:{}&1.051495355173&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.125428466373&{}:{}&-0.036350136674&{}:{}&0.910921670301&. \end{alignedat} \]
5b (303)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5b}}&{}\approx{}&-0.006651931837&{}:{}&0.027551890820&{}:{}&0.979100041017&,\\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.018151990823&{}:{}&0.026873011709&{}:{}&0.954974997469&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.007065770353&{}:{}&-0.032947298320&{}:{}&1.040013068674&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.103411961942&{}:{}&0.428325959222&{}:{}&0.675086002720&. \end{alignedat} \]
5b (303)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5b}}&{}\approx{}&0.028783583756&{}:{}&-0.032469068711&{}:{}&1.003685484955&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&-1.155831533393&{}:{}&-5.389578833482&{}:{}&7.545410366875&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.207737056431&{}:{}&0.307543145230&{}:{}&0.484719798339&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.019251774013&{}:{}&-0.032086290022&{}:{}&1.012834516009&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.035237917714&{}:{}&-0.003281043866&{}:{}&0.968043126152&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.012297441176&{}:{}&-0.063792796349&{}:{}&1.051495355173&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.125428466373&{}:{}&-0.036350136674&{}:{}&0.910921670301&, \end{alignedat} \]
5b (303)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5b}}&{}\approx{}&0.236893889655&{}:{}&0.032927836551&{}:{}&0.730178273794&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.043149748252&{}:{}&0.956850251748&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.244959888836&{}:{}&0.000000000000&{}:{}&0.755040111164&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.877964473009&{}:{}&0.122035526991&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5b}}&{}\approx{}&0.069300614852&{}:{}&-0.048163321605&{}:{}&0.978862706753&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.035237917714&{}:{}&-0.003281043866&{}:{}&0.968043126152&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.012297441176&{}:{}&-0.063792796349&{}:{}&1.051495355173&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.125428466373&{}:{}&-0.036350136674&{}:{}&0.910921670301&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.043149748252&{}:{}&0.956850251748&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.244959888836&{}:{}&0.000000000000&{}:{}&0.755040111164&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.877964473009&{}:{}&0.122035526991&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)