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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\(\mathbf{7a}\) \((233)\)

Malfatti circles

7a (233)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&, \\ P_{\mathbf{7a}}&{}\approx{}&1.001693457405&{}:{}&-0.000895289751&{}:{}&-0.000798167654&, \\ P^-_{\mathbf{7a}}&{}\approx{}&0.977235102325&{}:{}&0.010457827868&{}:{}&0.012307069807&, \\ P^+_{\mathbf{7a}}&{}\approx{}&1.027059016242&{}:{}&-0.012669514632&{}:{}&-0.014389501610&, \\ Q_{\mathbf{7a}}&{}\approx{}&0.884531603698&{}:{}&0.067232897738&{}:{}&0.048235498564&, \\ I^\prime_{\mathbf{7a}}&{}\approx{}&1.086089827246&{}:{}&-0.042732711162&{}:{}&-0.043357116083&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{7a}}\)
\(P^-_{\mathbf{7a}}\)
\(P^+_{\mathbf{7a}}\)
\(Q_{\mathbf{7a}}\)
\(I^\prime_{\mathbf{7a}}\)
7a (233)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7a}}&{}\approx{}&1.025638005274&{}:{}&-0.011898639656&{}:{}&-0.013739365617&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.673032607476&{}:{}&2.673032607476&{}:{}&-3.346065214951&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.870937466894&{}:{}&-1.508507942876&{}:{}&1.637570475982&, \\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.145537674257&{}:{}&-0.076942229376&{}:{}&-0.068595444881&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.024427325195&{}:{}&-0.023611042779&{}:{}&-0.000816282416&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.028285418516&{}:{}&-0.000919057012&{}:{}&-0.027366361504&, \\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.223316181277&{}:{}&0.356796055057&{}:{}&0.419887763666&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.999008426431&{}:{}&-0.011589704378&{}:{}&0.012581277947&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.002701755568&{}:{}&0.010730357862&{}:{}&-0.013432113430&, \\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.582262332300&{}:{}&0.417737667700&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.948287736084&{}:{}&0.000000000000&{}:{}&0.051712263916&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.929359733804&{}:{}&0.070640266196&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7a}}}{B^\prime_{\mathbf{7a}}}{C^\prime_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^*_{\mathbf{7a}}}{B^*_{\mathbf{7a}}}{C^*_{\mathbf{7a}}}\)
7a (233)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7a}}}}&{}\approx{}&-0.018768322465&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7a}}}}&{}\approx{}&-4.570810086343&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7a}}}}&{}\approx{}&-2.379445409771&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.025638005274&{}:{}&-0.011898639656&{}:{}&-0.013739365617&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.673032607476&{}:{}&2.673032607476&{}:{}&-3.346065214951&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.870937466894&{}:{}&-1.508507942876&{}:{}&1.637570475982&. \end{alignedat} \]
7a (233)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7a}}&{}\approx{}&1.001693457405&{}:{}&-0.000895289751&{}:{}&-0.000798167654&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.145537674257&{}:{}&-0.076942229376&{}:{}&-0.068595444881&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.024427325195&{}:{}&-0.023611042779&{}:{}&-0.000816282416&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.028285418516&{}:{}&-0.000919057012&{}:{}&-0.027366361504&. \end{alignedat} \]
7a (233)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7a}}&{}\approx{}&0.977235102325&{}:{}&0.010457827868&{}:{}&0.012307069807&,\\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.223316181277&{}:{}&0.356796055057&{}:{}&0.419887763666&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.999008426431&{}:{}&-0.011589704378&{}:{}&0.012581277947&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.002701755568&{}:{}&0.010730357862&{}:{}&-0.013432113430&. \end{alignedat} \]
7a (233)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7a}}&{}\approx{}&1.027059016242&{}:{}&-0.012669514632&{}:{}&-0.014389501610&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.025638005274&{}:{}&-0.011898639656&{}:{}&-0.013739365617&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.673032607476&{}:{}&2.673032607476&{}:{}&-3.346065214951&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.870937466894&{}:{}&-1.508507942876&{}:{}&1.637570475982&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.145537674257&{}:{}&-0.076942229376&{}:{}&-0.068595444881&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.024427325195&{}:{}&-0.023611042779&{}:{}&-0.000816282416&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.028285418516&{}:{}&-0.000919057012&{}:{}&-0.027366361504&, \end{alignedat} \]
7a (233)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7a}}&{}\approx{}&0.884531603698&{}:{}&0.067232897738&{}:{}&0.048235498564&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.582262332300&{}:{}&0.417737667700&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.948287736084&{}:{}&0.000000000000&{}:{}&0.051712263916&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.929359733804&{}:{}&0.070640266196&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7a}}&{}\approx{}&1.086089827246&{}:{}&-0.042732711162&{}:{}&-0.043357116083&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.145537674257&{}:{}&-0.076942229376&{}:{}&-0.068595444881&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.024427325195&{}:{}&-0.023611042779&{}:{}&-0.000816282416&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.028285418516&{}:{}&-0.000919057012&{}:{}&-0.027366361504&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.582262332300&{}:{}&0.417737667700&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.948287736084&{}:{}&0.000000000000&{}:{}&0.051712263916&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.929359733804&{}:{}&0.070640266196&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Hiroyasu Kamo