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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\(\mathbf{6}\) \((220)\)

Malfatti circles

6 (220)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&, \\ P_{\mathbf{6}}&{}\approx{}&0.898605744758&{}:{}&0.100678229761&{}:{}&0.000716025481&, \\ P^-_{\mathbf{6}}&{}\approx{}&1.217239636808&{}:{}&-0.022340764865&{}:{}&-0.194898871943&, \\ P^+_{\mathbf{6}}&{}\approx{}&0.733273214847&{}:{}&0.164510239212&{}:{}&0.102216545942&, \\ Q_{\mathbf{6}}&{}\approx{}&0.695990335685&{}:{}&0.341963583313&{}:{}&-0.037953918998&, \\ I^\prime_{\mathbf{6}}&{}\approx{}&0.675475670842&{}:{}&0.297559085274&{}:{}&0.026965243884&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{6}}\)
\(P^-_{\mathbf{6}}\)
\(P^+_{\mathbf{6}}\)
\(Q_{\mathbf{6}}\)
\(I^\prime_{\mathbf{6}}\)
6 (220)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6}}&{}\approx{}&0.769002401230&{}:{}&0.107206358682&{}:{}&0.123791240088&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.360754110767&{}:{}&-0.082262332300&{}:{}&0.721508221533&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.346352670420&{}:{}&2.331951230074&{}:{}&-2.678303900494&, \\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.569035593729&{}:{}&0.427921023828&{}:{}&0.003043382443&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.794378219640&{}:{}&0.204988805276&{}:{}&0.000632975084&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.709215823296&{}:{}&0.079459311299&{}:{}&0.211324865405&, \\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&-0.821468465999&{}:{}&0.187318480668&{}:{}&1.634149985330&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.021143590653&{}:{}&0.142357274659&{}:{}&-0.163500865312&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.875179475926&{}:{}&-0.016062719530&{}:{}&0.140883243603&, \\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.124844448887&{}:{}&-0.124844448887&,\\B^*_{\mathbf{6}}&{}\approx{}&1.057677535825&{}:{}&0.000000000000&{}:{}&-0.057677535825&,\\C^*_{\mathbf{6}}&{}\approx{}&0.670540688701&{}:{}&0.329459311299&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6}}}{B^\prime_{\mathbf{6}}}{C^\prime_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6}}}{B^{\prime\prime}_{\mathbf{6}}}{C^{\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6}}}{B^{\prime\prime\prime}_{\mathbf{6}}}{C^{\prime\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^*_{\mathbf{6}}}{B^*_{\mathbf{6}}}{C^*_{\mathbf{6}}}\)
6 (220)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6}}}}&{}\approx{}&0.292893218813&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6}}}}&{}\approx{}&1.707106781187&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6}}}}&{}\approx{}&6.371009241334&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.769002401230&{}:{}&0.107206358682&{}:{}&0.123791240088&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.360754110767&{}:{}&-0.082262332300&{}:{}&0.721508221533&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.346352670420&{}:{}&2.331951230074&{}:{}&-2.678303900494&. \end{alignedat} \]
6 (220)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6}}&{}\approx{}&0.898605744758&{}:{}&0.100678229761&{}:{}&0.000716025481&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.569035593729&{}:{}&0.427921023828&{}:{}&0.003043382443&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.794378219640&{}:{}&0.204988805276&{}:{}&0.000632975084&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.709215823296&{}:{}&0.079459311299&{}:{}&0.211324865405&. \end{alignedat} \]
6 (220)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6}}&{}\approx{}&1.217239636808&{}:{}&-0.022340764865&{}:{}&-0.194898871943&,\\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&-0.821468465999&{}:{}&0.187318480668&{}:{}&1.634149985330&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.021143590653&{}:{}&0.142357274659&{}:{}&-0.163500865312&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.875179475926&{}:{}&-0.016062719530&{}:{}&0.140883243603&. \end{alignedat} \]
6 (220)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6}}&{}\approx{}&0.733273214847&{}:{}&0.164510239212&{}:{}&0.102216545942&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.769002401230&{}:{}&0.107206358682&{}:{}&0.123791240088&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.360754110767&{}:{}&-0.082262332300&{}:{}&0.721508221533&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.346352670420&{}:{}&2.331951230074&{}:{}&-2.678303900494&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.569035593729&{}:{}&0.427921023828&{}:{}&0.003043382443&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.794378219640&{}:{}&0.204988805276&{}:{}&0.000632975084&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.709215823296&{}:{}&0.079459311299&{}:{}&0.211324865405&, \end{alignedat} \]
6 (220)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6}}&{}\approx{}&0.695990335685&{}:{}&0.341963583313&{}:{}&-0.037953918998&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.124844448887&{}:{}&-0.124844448887&,\\B^*_{\mathbf{6}}&{}\approx{}&1.057677535825&{}:{}&0.000000000000&{}:{}&-0.057677535825&,\\C^*_{\mathbf{6}}&{}\approx{}&0.670540688701&{}:{}&0.329459311299&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6}}&{}\approx{}&0.675475670842&{}:{}&0.297559085274&{}:{}&0.026965243884&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.569035593729&{}:{}&0.427921023828&{}:{}&0.003043382443&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.794378219640&{}:{}&0.204988805276&{}:{}&0.000632975084&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.709215823296&{}:{}&0.079459311299&{}:{}&0.211324865405&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.124844448887&{}:{}&-0.124844448887&,\\B^*_{\mathbf{6}}&{}\approx{}&1.057677535825&{}:{}&0.000000000000&{}:{}&-0.057677535825&,\\C^*_{\mathbf{6}}&{}\approx{}&0.670540688701&{}:{}&0.329459311299&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)

Hiroyasu Kamo