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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4b (301)

Malfatti circles

4b (301)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&, \\ P_{\mathbf{4b}}&{}\approx{}&0.026914085426&{}:{}&-0.021214449173&{}:{}&0.994300363747&, \\ P^-_{\mathbf{4b}}&{}\approx{}&-0.004247439381&{}:{}&0.033798024842&{}:{}&0.970449414539&, \\ P^+_{\mathbf{4b}}&{}\approx{}&0.055718956027&{}:{}&-0.072066492046&{}:{}&1.016347536019&, \\ Q_{\mathbf{4b}}&{}\approx{}&-0.400727311069&{}:{}&0.139933250190&{}:{}&1.260794060880&, \\ I^\prime_{\mathbf{4b}}&{}\approx{}&0.118661767585&{}:{}&-0.138649144325&{}:{}&1.019987376740&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{4b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{4b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{4b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{4b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{4b}}\) Radical center of the Malfatti circles
4b (301)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4b}}&{}\approx{}&-0.965925826289&{}:{}&-12.707944308956&{}:{}&14.673870135245&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.247060318299&{}:{}&0.258819045103&{}:{}&0.494120636598&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.034074173711&{}:{}&-0.059018200093&{}:{}&1.024944026382&, \\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.059888604508&{}:{}&-0.020495564798&{}:{}&0.960606960290&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.029451445317&{}:{}&-0.117490749968&{}:{}&1.088039304651&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.207545428360&{}:{}&-0.163593221591&{}:{}&0.956047793232&, \\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.031126093212&{}:{}&0.032607525881&{}:{}&0.936266380907&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.004665866461&{}:{}&-0.061385221846&{}:{}&1.066051088307&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.045149215873&{}:{}&0.359264531564&{}:{}&0.685884684309&, \\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.099900422505&{}:{}&0.900099577495&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.465925826289&{}:{}&0.000000000000&{}:{}&1.465925826289&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.536566092485&{}:{}&-0.536566092485&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4b}}}{B^\prime_{\mathbf{4b}}}{C^\prime_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^*_{\mathbf{4b}}}{B^*_{\mathbf{4b}}}{C^*_{\mathbf{4b}}}\)
4b (301)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4b}}}}&{}\approx{}&9.302860893912&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4b}}}}&{}\approx{}&0.313259931069&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4b}}}}&{}\approx{}&0.043204321040&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-0.965925826289&{}:{}&-12.707944308956&{}:{}&14.673870135245&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.247060318299&{}:{}&0.258819045103&{}:{}&0.494120636598&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.034074173711&{}:{}&-0.059018200093&{}:{}&1.024944026382&. \end{alignedat} \]
4b (301)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4b}}&{}\approx{}&0.026914085426&{}:{}&-0.021214449173&{}:{}&0.994300363747&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.059888604508&{}:{}&-0.020495564798&{}:{}&0.960606960290&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.029451445317&{}:{}&-0.117490749968&{}:{}&1.088039304651&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.207545428360&{}:{}&-0.163593221591&{}:{}&0.956047793232&. \end{alignedat} \]
4b (301)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4b}}&{}\approx{}&-0.004247439381&{}:{}&0.033798024842&{}:{}&0.970449414539&,\\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.031126093212&{}:{}&0.032607525881&{}:{}&0.936266380907&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.004665866461&{}:{}&-0.061385221846&{}:{}&1.066051088307&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.045149215873&{}:{}&0.359264531564&{}:{}&0.685884684309&. \end{alignedat} \]
4b (301)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4b}}&{}\approx{}&0.055718956027&{}:{}&-0.072066492046&{}:{}&1.016347536019&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-0.965925826289&{}:{}&-12.707944308956&{}:{}&14.673870135245&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.247060318299&{}:{}&0.258819045103&{}:{}&0.494120636598&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.034074173711&{}:{}&-0.059018200093&{}:{}&1.024944026382&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.059888604508&{}:{}&-0.020495564798&{}:{}&0.960606960290&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.029451445317&{}:{}&-0.117490749968&{}:{}&1.088039304651&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.207545428360&{}:{}&-0.163593221591&{}:{}&0.956047793232&, \end{alignedat} \]
4b (301)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4b}}&{}\approx{}&-0.400727311069&{}:{}&0.139933250190&{}:{}&1.260794060880&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.099900422505&{}:{}&0.900099577495&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.465925826289&{}:{}&0.000000000000&{}:{}&1.465925826289&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.536566092485&{}:{}&-0.536566092485&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4b}}&{}\approx{}&0.118661767585&{}:{}&-0.138649144325&{}:{}&1.019987376740&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.059888604508&{}:{}&-0.020495564798&{}:{}&0.960606960290&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.029451445317&{}:{}&-0.117490749968&{}:{}&1.088039304651&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.207545428360&{}:{}&-0.163593221591&{}:{}&0.956047793232&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.099900422505&{}:{}&0.900099577495&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.465925826289&{}:{}&0.000000000000&{}:{}&1.465925826289&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.536566092485&{}:{}&-0.536566092485&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)