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]

\(\mathbf{7}\) \((222)\)

Malfatti circles

7 (222)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&, \\ P_{\mathbf{7}}&{}\approx{}&0.877880939556&{}:{}&0.098356258516&{}:{}&0.023762801928&, \\ P^-_{\mathbf{7}}&{}\approx{}&-0.618101828707&{}:{}&0.699098558295&{}:{}&0.919003270411&, \\ P^+_{\mathbf{7}}&{}\approx{}&0.449071984770&{}:{}&0.270553213674&{}:{}&0.280374801556&, \\ Q_{\mathbf{7}}&{}\approx{}&0.552739331730&{}:{}&0.271579521762&{}:{}&0.175681146508&, \\ I^\prime_{\mathbf{7}}&{}\approx{}&0.597660450519&{}:{}&0.263280092293&{}:{}&0.139059457188&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{7}}\)
\(P^-_{\mathbf{7}}\)
\(P^+_{\mathbf{7}}\)
\(Q_{\mathbf{7}}\)
\(I^\prime_{\mathbf{7}}\)
7 (222)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7}}&{}\approx{}&0.258819045103&{}:{}&0.343983278277&{}:{}&0.397197676620&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.655308608763&{}:{}&-0.965925826289&{}:{}&1.310617217526&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.258819045103&{}:{}&2.180338543653&{}:{}&-2.439157588755&, \\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.861922615730&{}:{}&0.111209297329&{}:{}&0.026868086941&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.395086645885&{}:{}&0.594219004474&{}:{}&0.010694349641&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.351033215403&{}:{}&0.039329152879&{}:{}&0.609637631718&, \\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.414652684042&{}:{}&0.611198649156&{}:{}&0.803454034886&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&1.187318480668&{}:{}&1.578004830277&{}:{}&-1.765323310945&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.712447143398&{}:{}&-0.805806984673&{}:{}&1.093359841275&, \\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.607206358682&{}:{}&0.392793641318&,\\B^*_{\mathbf{7}}&{}\approx{}&0.758819045103&{}:{}&0.000000000000&{}:{}&0.241180954897&,\\C^*_{\mathbf{7}}&{}\approx{}&0.670540688701&{}:{}&0.329459311299&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7}}}{B^\prime_{\mathbf{7}}}{C^\prime_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7}}}{B^{\prime\prime}_{\mathbf{7}}}{C^{\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7}}}{B^{\prime\prime\prime}_{\mathbf{7}}}{C^{\prime\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^*_{\mathbf{7}}}{B^*_{\mathbf{7}}}{C^*_{\mathbf{7}}}\)
7 (222)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7}}}}&{}\approx{}&0.939779793208&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7}}}}&{}\approx{}&3.100953631304&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7}}}}&{}\approx{}&5.956795678960&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.258819045103&{}:{}&0.343983278277&{}:{}&0.397197676620&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.655308608763&{}:{}&-0.965925826289&{}:{}&1.310617217526&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.258819045103&{}:{}&2.180338543653&{}:{}&-2.439157588755&. \end{alignedat} \]
7 (222)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7}}&{}\approx{}&0.877880939556&{}:{}&0.098356258516&{}:{}&0.023762801928&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.861922615730&{}:{}&0.111209297329&{}:{}&0.026868086941&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.395086645885&{}:{}&0.594219004474&{}:{}&0.010694349641&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.351033215403&{}:{}&0.039329152879&{}:{}&0.609637631718&. \end{alignedat} \]
7 (222)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7}}&{}\approx{}&-0.618101828707&{}:{}&0.699098558295&{}:{}&0.919003270411&,\\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.414652684042&{}:{}&0.611198649156&{}:{}&0.803454034886&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&1.187318480668&{}:{}&1.578004830277&{}:{}&-1.765323310945&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.712447143398&{}:{}&-0.805806984673&{}:{}&1.093359841275&. \end{alignedat} \]
7 (222)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7}}&{}\approx{}&0.449071984770&{}:{}&0.270553213674&{}:{}&0.280374801556&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.258819045103&{}:{}&0.343983278277&{}:{}&0.397197676620&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.655308608763&{}:{}&-0.965925826289&{}:{}&1.310617217526&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.258819045103&{}:{}&2.180338543653&{}:{}&-2.439157588755&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.861922615730&{}:{}&0.111209297329&{}:{}&0.026868086941&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.395086645885&{}:{}&0.594219004474&{}:{}&0.010694349641&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.351033215403&{}:{}&0.039329152879&{}:{}&0.609637631718&, \end{alignedat} \]
7 (222)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7}}&{}\approx{}&0.552739331730&{}:{}&0.271579521762&{}:{}&0.175681146508&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.607206358682&{}:{}&0.392793641318&,\\B^*_{\mathbf{7}}&{}\approx{}&0.758819045103&{}:{}&0.000000000000&{}:{}&0.241180954897&,\\C^*_{\mathbf{7}}&{}\approx{}&0.670540688701&{}:{}&0.329459311299&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7}}&{}\approx{}&0.597660450519&{}:{}&0.263280092293&{}:{}&0.139059457188&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.861922615730&{}:{}&0.111209297329&{}:{}&0.026868086941&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.395086645885&{}:{}&0.594219004474&{}:{}&0.010694349641&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.351033215403&{}:{}&0.039329152879&{}:{}&0.609637631718&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.607206358682&{}:{}&0.392793641318&,\\B^*_{\mathbf{7}}&{}\approx{}&0.758819045103&{}:{}&0.000000000000&{}:{}&0.241180954897&,\\C^*_{\mathbf{7}}&{}\approx{}&0.670540688701&{}:{}&0.329459311299&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Hiroyasu Kamo