Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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1b(103)

Malfatti circles

1b (103)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&, \\ P_{\mathbf{1b}}&{}\approx{}&0.868845561449&{}:{}&-0.251798142965&{}:{}&0.382952581516&, \\ P^-_{\mathbf{1b}}&{}\approx{}&0.920740953615&{}:{}&-0.107372367136&{}:{}&0.186631413522&, \\ P^+_{\mathbf{1b}}&{}\approx{}&0.831404646193&{}:{}&-0.355996859032&{}:{}&0.524592212839&, \\ Q_{\mathbf{1b}}&{}\approx{}&1.089681915053&{}:{}&0.151463881335&{}:{}&-0.241145796388&, \\ I^\prime_{\mathbf{1b}}&{}\approx{}&0.758075783567&{}:{}&-0.526856043671&{}:{}&0.768780260103&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{1b}}\)
\(P^-_{\mathbf{1b}}\)
\(P^+_{\mathbf{1b}}\)
\(Q_{\mathbf{1b}}\)
\(I^\prime_{\mathbf{1b}}\)
1b (103)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1b}}&{}\approx{}&0.922629435690&{}:{}&-0.193426410776&{}:{}&0.270796975086&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.415583153339&{}:{}&-0.385277177798&{}:{}&0.969694024458&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.058482258555&{}:{}&-1.764137097591&{}:{}&1.705654839036&, \\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.596835650867&{}:{}&-0.774019053740&{}:{}&1.177183402873&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.936053022617&{}:{}&-0.348627981105&{}:{}&0.412574958488&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.811942135629&{}:{}&-0.235307091407&{}:{}&0.423364955778&, \\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.593704503647&{}:{}&-0.550409211184&{}:{}&0.956704707537&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.006998566976&{}:{}&-0.211114138496&{}:{}&0.204115571520&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.849696721928&{}:{}&-0.099087531648&{}:{}&0.249390809721&, \\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.688901059317&{}:{}&2.688901059317&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.284190373377&{}:{}&0.000000000000&{}:{}&-0.284190373377&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.877964473009&{}:{}&0.122035526991&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1b}}}{B^\prime_{\mathbf{1b}}}{C^\prime_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^*_{\mathbf{1b}}}{B^*_{\mathbf{1b}}}{C^*_{\mathbf{1b}}}\)
1b (103)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1b}}}}&{}\approx{}&0.193426410776&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1b}}}}&{}\approx{}&0.692638588899&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1b}}}}&{}\approx{}&1.764137097591&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.922629435690&{}:{}&-0.193426410776&{}:{}&0.270796975086&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.415583153339&{}:{}&-0.385277177798&{}:{}&0.969694024458&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.058482258555&{}:{}&-1.764137097591&{}:{}&1.705654839036&. \end{alignedat} \]
1b (103)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1b}}&{}\approx{}&0.868845561449&{}:{}&-0.251798142965&{}:{}&0.382952581516&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.596835650867&{}:{}&-0.774019053740&{}:{}&1.177183402873&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.936053022617&{}:{}&-0.348627981105&{}:{}&0.412574958488&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.811942135629&{}:{}&-0.235307091407&{}:{}&0.423364955778&. \end{alignedat} \]
1b (103)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1b}}&{}\approx{}&0.920740953615&{}:{}&-0.107372367136&{}:{}&0.186631413522&,\\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.593704503647&{}:{}&-0.550409211184&{}:{}&0.956704707537&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.006998566976&{}:{}&-0.211114138496&{}:{}&0.204115571520&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.849696721928&{}:{}&-0.099087531648&{}:{}&0.249390809721&. \end{alignedat} \]
1b (103)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1b}}&{}\approx{}&0.831404646193&{}:{}&-0.355996859032&{}:{}&0.524592212839&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.922629435690&{}:{}&-0.193426410776&{}:{}&0.270796975086&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.415583153339&{}:{}&-0.385277177798&{}:{}&0.969694024458&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.058482258555&{}:{}&-1.764137097591&{}:{}&1.705654839036&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.596835650867&{}:{}&-0.774019053740&{}:{}&1.177183402873&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.936053022617&{}:{}&-0.348627981105&{}:{}&0.412574958488&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.811942135629&{}:{}&-0.235307091407&{}:{}&0.423364955778&, \end{alignedat} \]
1b (103)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1b}}&{}\approx{}&1.089681915053&{}:{}&0.151463881335&{}:{}&-0.241145796388&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.688901059317&{}:{}&2.688901059317&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.284190373377&{}:{}&0.000000000000&{}:{}&-0.284190373377&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.877964473009&{}:{}&0.122035526991&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1b}}&{}\approx{}&0.758075783567&{}:{}&-0.526856043671&{}:{}&0.768780260103&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.596835650867&{}:{}&-0.774019053740&{}:{}&1.177183402873&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.936053022617&{}:{}&-0.348627981105&{}:{}&0.412574958488&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.811942135629&{}:{}&-0.235307091407&{}:{}&0.423364955778&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.688901059317&{}:{}&2.688901059317&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.284190373377&{}:{}&0.000000000000&{}:{}&-0.284190373377&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.877964473009&{}:{}&0.122035526991&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)