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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6a (231)

Malfatti circles

6a (231)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&, \\ P_{\mathbf{6a}}&{}\approx{}&1.008919660857&{}:{}&-0.008115735232&{}:{}&-0.000803925625&, \\ P^-_{\mathbf{6a}}&{}\approx{}&0.937217010609&{}:{}&0.025368932372&{}:{}&0.037414057019&, \\ P^+_{\mathbf{6a}}&{}\approx{}&1.088971647769&{}:{}&-0.045499488322&{}:{}&-0.043472159447&, \\ Q_{\mathbf{6a}}&{}\approx{}&1.209916326228&{}:{}&-0.275895785834&{}:{}&0.065979459606&, \\ I^\prime_{\mathbf{6a}}&{}\approx{}&1.187587493965&{}:{}&-0.140178552694&{}:{}&-0.047408941271&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{6a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{6a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{6a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{6a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{6a}}\) Radical center of the Malfatti circles
6a (231)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6a}}&{}\approx{}&1.082262332300&{}:{}&-0.038178081285&{}:{}&-0.044084251014&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&3.039058011260&{}:{}&4.039058011260&{}:{}&-6.078116022520&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.814313139868&{}:{}&-1.410431731523&{}:{}&1.596118591655&, \\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.284457050376&{}:{}&-0.258819045103&{}:{}&-0.025638005274&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.075200536943&{}:{}&-0.074343797505&{}:{}&-0.000856739439&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.096352670420&{}:{}&-0.008819045103&{}:{}&-0.087533625317&, \\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.233147571756&{}:{}&0.309864623847&{}:{}&0.456987804396&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.995377271606&{}:{}&-0.035113108209&{}:{}&0.039735836603&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.013854380176&{}:{}&0.027443380685&{}:{}&-0.041297760861&, \\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.314313139868&{}:{}&-0.314313139868&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.948287736084&{}:{}&0.000000000000&{}:{}&0.051712263916&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.295385137588&{}:{}&-0.295385137588&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6a}}}{B^\prime_{\mathbf{6a}}}{C^\prime_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^*_{\mathbf{6a}}}{B^*_{\mathbf{6a}}}{C^*_{\mathbf{6a}}}\)
6a (231)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6a}}}}&{}\approx{}&-0.060220206792&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6a}}}}&{}\approx{}&-8.302860893912&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6a}}}}&{}\approx{}&-2.224744871392&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.082262332300&{}:{}&-0.038178081285&{}:{}&-0.044084251014&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&3.039058011260&{}:{}&4.039058011260&{}:{}&-6.078116022520&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.814313139868&{}:{}&-1.410431731523&{}:{}&1.596118591655&. \end{alignedat} \]
6a (231)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6a}}&{}\approx{}&1.008919660857&{}:{}&-0.008115735232&{}:{}&-0.000803925625&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.284457050376&{}:{}&-0.258819045103&{}:{}&-0.025638005274&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.075200536943&{}:{}&-0.074343797505&{}:{}&-0.000856739439&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.096352670420&{}:{}&-0.008819045103&{}:{}&-0.087533625317&. \end{alignedat} \]
6a (231)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6a}}&{}\approx{}&0.937217010609&{}:{}&0.025368932372&{}:{}&0.037414057019&,\\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.233147571756&{}:{}&0.309864623847&{}:{}&0.456987804396&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.995377271606&{}:{}&-0.035113108209&{}:{}&0.039735836603&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.013854380176&{}:{}&0.027443380685&{}:{}&-0.041297760861&. \end{alignedat} \]
6a (231)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6a}}&{}\approx{}&1.088971647769&{}:{}&-0.045499488322&{}:{}&-0.043472159447&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.082262332300&{}:{}&-0.038178081285&{}:{}&-0.044084251014&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&3.039058011260&{}:{}&4.039058011260&{}:{}&-6.078116022520&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.814313139868&{}:{}&-1.410431731523&{}:{}&1.596118591655&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.284457050376&{}:{}&-0.258819045103&{}:{}&-0.025638005274&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.075200536943&{}:{}&-0.074343797505&{}:{}&-0.000856739439&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.096352670420&{}:{}&-0.008819045103&{}:{}&-0.087533625317&, \end{alignedat} \]
6a (231)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6a}}&{}\approx{}&1.209916326228&{}:{}&-0.275895785834&{}:{}&0.065979459606&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.314313139868&{}:{}&-0.314313139868&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.948287736084&{}:{}&0.000000000000&{}:{}&0.051712263916&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.295385137588&{}:{}&-0.295385137588&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6a}}&{}\approx{}&1.187587493965&{}:{}&-0.140178552694&{}:{}&-0.047408941271&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.284457050376&{}:{}&-0.258819045103&{}:{}&-0.025638005274&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.075200536943&{}:{}&-0.074343797505&{}:{}&-0.000856739439&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.096352670420&{}:{}&-0.008819045103&{}:{}&-0.087533625317&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.314313139868&{}:{}&-0.314313139868&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.948287736084&{}:{}&0.000000000000&{}:{}&0.051712263916&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.295385137588&{}:{}&-0.295385137588&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)