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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\(\mathbf{4c}\) \((310)\)

Malfatti circles

4c (310)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&, \\ P_{\mathbf{4c}}&{}\approx{}&0.091818159149&{}:{}&1.008035460348&{}:{}&-0.099853619498&, \\ P^-_{\mathbf{4c}}&{}\approx{}&0.025368932372&{}:{}&0.937217010609&{}:{}&0.037414057019&, \\ P^+_{\mathbf{4c}}&{}\approx{}&0.151991383001&{}:{}&1.072165242374&{}:{}&-0.224156625375&, \\ Q_{\mathbf{4c}}&{}\approx{}&-0.556462532683&{}:{}&1.256076476151&{}:{}&0.300386056532&, \\ I^\prime_{\mathbf{4c}}&{}\approx{}&0.257342043172&{}:{}&1.122185551259&{}:{}&-0.379527594431&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{4c}}\)
\(P^-_{\mathbf{4c}}\)
\(P^+_{\mathbf{4c}}\)
\(Q_{\mathbf{4c}}\)
\(I^\prime_{\mathbf{4c}}\)
4c (310)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4c}}&{}\approx{}&4.039058011260&{}:{}&19.644779799083&{}:{}&-22.683837810344&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.082262332300&{}:{}&1.082262332300&{}:{}&-0.164524664599&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.307007203691&{}:{}&0.531752075083&{}:{}&0.161240721226&, \\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.129756511997&{}:{}&0.965925826289&{}:{}&-0.095682338286&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.405365324582&{}:{}&1.035475520556&{}:{}&-0.440840845138&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.110754110767&{}:{}&1.215925826289&{}:{}&-0.326679937056&, \\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.068113238382&{}:{}&0.896113569492&{}:{}&0.035773192126&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.136267600782&{}:{}&0.662765180311&{}:{}&0.200967218906&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.030933536812&{}:{}&1.142792943468&{}:{}&-0.173726480280&, \\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.807007203691&{}:{}&0.192992796309&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.173032607476&{}:{}&0.000000000000&{}:{}&-1.173032607476&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.795385137588&{}:{}&1.795385137588&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4c}}}{B^\prime_{\mathbf{4c}}}{C^\prime_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^*_{\mathbf{4c}}}{B^*_{\mathbf{4c}}}{C^*_{\mathbf{4c}}}\)
4c (310)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4c}}}}&{}\approx{}&8.302860893912&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4c}}}}&{}\approx{}&0.060220206792&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4c}}}}&{}\approx{}&0.224744871392&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&4.039058011260&{}:{}&19.644779799083&{}:{}&-22.683837810344&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.082262332300&{}:{}&1.082262332300&{}:{}&-0.164524664599&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.307007203691&{}:{}&0.531752075083&{}:{}&0.161240721226&. \end{alignedat} \]
4c (310)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4c}}&{}\approx{}&0.091818159149&{}:{}&1.008035460348&{}:{}&-0.099853619498&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.129756511997&{}:{}&0.965925826289&{}:{}&-0.095682338286&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.405365324582&{}:{}&1.035475520556&{}:{}&-0.440840845138&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.110754110767&{}:{}&1.215925826289&{}:{}&-0.326679937056&. \end{alignedat} \]
4c (310)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4c}}&{}\approx{}&0.025368932372&{}:{}&0.937217010609&{}:{}&0.037414057019&,\\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.068113238382&{}:{}&0.896113569492&{}:{}&0.035773192126&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.136267600782&{}:{}&0.662765180311&{}:{}&0.200967218906&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.030933536812&{}:{}&1.142792943468&{}:{}&-0.173726480280&. \end{alignedat} \]
4c (310)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4c}}&{}\approx{}&0.151991383001&{}:{}&1.072165242374&{}:{}&-0.224156625375&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&4.039058011260&{}:{}&19.644779799083&{}:{}&-22.683837810344&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.082262332300&{}:{}&1.082262332300&{}:{}&-0.164524664599&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.307007203691&{}:{}&0.531752075083&{}:{}&0.161240721226&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.129756511997&{}:{}&0.965925826289&{}:{}&-0.095682338286&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.405365324582&{}:{}&1.035475520556&{}:{}&-0.440840845138&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.110754110767&{}:{}&1.215925826289&{}:{}&-0.326679937056&, \end{alignedat} \]
4c (310)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4c}}&{}\approx{}&-0.556462532683&{}:{}&1.256076476151&{}:{}&0.300386056532&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.807007203691&{}:{}&0.192992796309&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.173032607476&{}:{}&0.000000000000&{}:{}&-1.173032607476&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.795385137588&{}:{}&1.795385137588&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4c}}&{}\approx{}&0.257342043172&{}:{}&1.122185551259&{}:{}&-0.379527594431&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.129756511997&{}:{}&0.965925826289&{}:{}&-0.095682338286&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.405365324582&{}:{}&1.035475520556&{}:{}&-0.440840845138&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.110754110767&{}:{}&1.215925826289&{}:{}&-0.326679937056&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.807007203691&{}:{}&0.192992796309&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.173032607476&{}:{}&0.000000000000&{}:{}&-1.173032607476&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.795385137588&{}:{}&1.795385137588&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Hiroyasu Kamo