Derousseau's Generalization of the Malfatti circles

Half of an Equilateral Triangle

\(A=30\degree\), \(B=60\degree\), \(C=90\degree\).


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[Lob & Richmond]

\(\mathbf{3c}\) \((132)\)

Malfatti circles

3c (132)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&, \\ P_{\mathbf{3c}}&{}\approx{}&-0.040056258777&{}:{}&-0.439761912346&{}:{}&1.479818171123&, \\ P^-_{\mathbf{3c}}&{}\approx{}&0.117315613893&{}:{}&-0.125731784503&{}:{}&1.008416170609&, \\ P^+_{\mathbf{3c}}&{}\approx{}&-0.242814452066&{}:{}&-0.844358872907&{}:{}&2.087173324972&, \\ Q_{\mathbf{3c}}&{}\approx{}&0.529377314879&{}:{}&-1.194938298938&{}:{}&1.665560984059&, \\ I^\prime_{\mathbf{3c}}&{}\approx{}&-0.309111650864&{}:{}&-1.347936093342&{}:{}&2.657047744207&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{3c}}\)
\(P^-_{\mathbf{3c}}\)
\(P^+_{\mathbf{3c}}\)
\(Q_{\mathbf{3c}}\)
\(I^\prime_{\mathbf{3c}}\)
3c (132)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3c}}&{}\approx{}&0.741180954897&{}:{}&-1.673032607476&{}:{}&1.931851652578&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-5.629828286436&{}:{}&-4.629828286436&{}:{}&11.259656572872&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.258819045103&{}:{}&-0.448287736084&{}:{}&1.707106781187&, \\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.494886753833&{}:{}&-0.632075670964&{}:{}&2.126962424797&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.047494179697&{}:{}&-0.707106781187&{}:{}&1.754600960884&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.191831746995&{}:{}&-2.106045309368&{}:{}&3.297877056363&, \\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.209495183861&{}:{}&-0.172283536684&{}:{}&1.381778720544&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.136267600782&{}:{}&-0.307590390640&{}:{}&1.171322789857&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.394522265867&{}:{}&-0.422825119925&{}:{}&1.028302854058&, \\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-2.539058011260&{}:{}&3.539058011260&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.241180954897&{}:{}&0.000000000000&{}:{}&0.758819045103&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.795385137588&{}:{}&1.795385137588&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3c}}}{B^\prime_{\mathbf{3c}}}{C^\prime_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^*_{\mathbf{3c}}}{B^*_{\mathbf{3c}}}{C^*_{\mathbf{3c}}}\)
3c (132)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.707106781187&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3c}}}}&{}\approx{}&-4.121320343560&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.189468690982&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.741180954897&{}:{}&-1.673032607476&{}:{}&1.931851652578&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-5.629828286436&{}:{}&-4.629828286436&{}:{}&11.259656572872&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.258819045103&{}:{}&-0.448287736084&{}:{}&1.707106781187&. \end{alignedat} \]
3c (132)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3c}}&{}\approx{}&-0.040056258777&{}:{}&-0.439761912346&{}:{}&1.479818171123&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.494886753833&{}:{}&-0.632075670964&{}:{}&2.126962424797&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.047494179697&{}:{}&-0.707106781187&{}:{}&1.754600960884&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.191831746995&{}:{}&-2.106045309368&{}:{}&3.297877056363&. \end{alignedat} \]
3c (132)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3c}}&{}\approx{}&0.117315613893&{}:{}&-0.125731784503&{}:{}&1.008416170609&,\\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.209495183861&{}:{}&-0.172283536684&{}:{}&1.381778720544&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.136267600782&{}:{}&-0.307590390640&{}:{}&1.171322789857&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.394522265867&{}:{}&-0.422825119925&{}:{}&1.028302854058&. \end{alignedat} \]
3c (132)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3c}}&{}\approx{}&-0.242814452066&{}:{}&-0.844358872907&{}:{}&2.087173324972&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.741180954897&{}:{}&-1.673032607476&{}:{}&1.931851652578&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-5.629828286436&{}:{}&-4.629828286436&{}:{}&11.259656572872&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.258819045103&{}:{}&-0.448287736084&{}:{}&1.707106781187&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.494886753833&{}:{}&-0.632075670964&{}:{}&2.126962424797&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.047494179697&{}:{}&-0.707106781187&{}:{}&1.754600960884&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.191831746995&{}:{}&-2.106045309368&{}:{}&3.297877056363&, \end{alignedat} \]
3c (132)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3c}}&{}\approx{}&0.529377314879&{}:{}&-1.194938298938&{}:{}&1.665560984059&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-2.539058011260&{}:{}&3.539058011260&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.241180954897&{}:{}&0.000000000000&{}:{}&0.758819045103&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.795385137588&{}:{}&1.795385137588&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3c}}&{}\approx{}&-0.309111650864&{}:{}&-1.347936093342&{}:{}&2.657047744207&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.494886753833&{}:{}&-0.632075670964&{}:{}&2.126962424797&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.047494179697&{}:{}&-0.707106781187&{}:{}&1.754600960884&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.191831746995&{}:{}&-2.106045309368&{}:{}&3.297877056363&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-2.539058011260&{}:{}&3.539058011260&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.241180954897&{}:{}&0.000000000000&{}:{}&0.758819045103&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.795385137588&{}:{}&1.795385137588&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Hiroyasu Kamo