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]

\(\mathbf{1}\) \((002)\)

Malfatti circles

1 (002)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&, \\ P_{\mathbf{1}}&{}\approx{}&0.010749591380&{}:{}&0.020665776851&{}:{}&0.968584631769&, \\ P^-_{\mathbf{1}}&{}\approx{}&-0.009447883861&{}:{}&-0.014111154395&{}:{}&1.023559038256&, \\ P^+_{\mathbf{1}}&{}\approx{}&0.027561270663&{}:{}&0.049612890664&{}:{}&0.922825838673&, \\ Q_{\mathbf{1}}&{}\approx{}&-0.065348814526&{}:{}&-0.133002885622&{}:{}&1.198351700147&, \\ I^\prime_{\mathbf{1}}&{}\approx{}&0.061542400480&{}:{}&0.112301261693&{}:{}&0.826156337828&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{1}}\)
\(P^-_{\mathbf{1}}\)
\(P^+_{\mathbf{1}}\)
\(Q_{\mathbf{1}}\)
\(I^\prime_{\mathbf{1}}\)
1 (002)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1}}&{}\approx{}&-0.135027805015&{}:{}&0.526768237534&{}:{}&0.608259567481&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.427921023828&{}:{}&-0.283763071485&{}:{}&0.855842047657&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.019672733364&{}:{}&0.034074173711&{}:{}&0.946253092925&, \\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.037616146640&{}:{}&0.020104525391&{}:{}&0.942279327969&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.010273848914&{}:{}&0.064007971121&{}:{}&0.925718179965&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.098848728967&{}:{}&0.190033807112&{}:{}&0.711117463921&, \\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.020645542240&{}:{}&-0.013690475934&{}:{}&0.993044933694&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.008989687726&{}:{}&0.035070420932&{}:{}&0.973919266794&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.497235293230&{}:{}&-0.742659848096&{}:{}&2.239895141326&, \\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.124844448887&{}:{}&1.124844448887&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.057677535825&{}:{}&0.000000000000&{}:{}&1.057677535825&,\\C^*_{\mathbf{1}}&{}\approx{}&0.329459311299&{}:{}&0.670540688701&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1}}}{B^\prime_{\mathbf{1}}}{C^\prime_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1}}}{B^{\prime\prime}_{\mathbf{1}}}{C^{\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1}}}{B^{\prime\prime\prime}_{\mathbf{1}}}{C^{\prime\prime\prime}_{\mathbf{1}}}\)
\(\triangle{A^*_{\mathbf{1}}}{B^*_{\mathbf{1}}}{C^*_{\mathbf{1}}}\)
1 (002)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1}}}}&{}\approx{}&1.439157588755&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1}}}}&{}\approx{}&2.024944026382&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1}}}}&{}\approx{}&0.093092373804&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.135027805015&{}:{}&0.526768237534&{}:{}&0.608259567481&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.427921023828&{}:{}&-0.283763071485&{}:{}&0.855842047657&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.019672733364&{}:{}&0.034074173711&{}:{}&0.946253092925&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1}}&{}\approx{}&0.010749591380&{}:{}&0.020665776851&{}:{}&0.968584631769&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.037616146640&{}:{}&0.020104525391&{}:{}&0.942279327969&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.010273848914&{}:{}&0.064007971121&{}:{}&0.925718179965&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.098848728967&{}:{}&0.190033807112&{}:{}&0.711117463921&. \end{alignedat} \]
1 (002)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1}}&{}\approx{}&-0.009447883861&{}:{}&-0.014111154395&{}:{}&1.023559038256&,\\ A^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&0.020645542240&{}:{}&-0.013690475934&{}:{}&0.993044933694&,\\B^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.008989687726&{}:{}&0.035070420932&{}:{}&0.973919266794&,\\C^{\prime\prime\prime}_{\mathbf{1}}&{}\approx{}&-0.497235293230&{}:{}&-0.742659848096&{}:{}&2.239895141326&. \end{alignedat} \]
1 (002)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1}}&{}\approx{}&0.027561270663&{}:{}&0.049612890664&{}:{}&0.922825838673&,\\ A^\prime_{\mathbf{1}}&{}\approx{}&-0.135027805015&{}:{}&0.526768237534&{}:{}&0.608259567481&,\\B^\prime_{\mathbf{1}}&{}\approx{}&0.427921023828&{}:{}&-0.283763071485&{}:{}&0.855842047657&,\\C^\prime_{\mathbf{1}}&{}\approx{}&0.019672733364&{}:{}&0.034074173711&{}:{}&0.946253092925&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.037616146640&{}:{}&0.020104525391&{}:{}&0.942279327969&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.010273848914&{}:{}&0.064007971121&{}:{}&0.925718179965&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.098848728967&{}:{}&0.190033807112&{}:{}&0.711117463921&, \end{alignedat} \]
1 (002)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1}}&{}\approx{}&-0.065348814526&{}:{}&-0.133002885622&{}:{}&1.198351700147&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.124844448887&{}:{}&1.124844448887&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.057677535825&{}:{}&0.000000000000&{}:{}&1.057677535825&,\\C^*_{\mathbf{1}}&{}\approx{}&0.329459311299&{}:{}&0.670540688701&{}:{}&0.000000000000&. \end{alignedat} \]
1 (002)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1}}&{}\approx{}&0.061542400480&{}:{}&0.112301261693&{}:{}&0.826156337828&,\\ A^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.037616146640&{}:{}&0.020104525391&{}:{}&0.942279327969&,\\B^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.010273848914&{}:{}&0.064007971121&{}:{}&0.925718179965&,\\C^{\prime\prime}_{\mathbf{1}}&{}\approx{}&0.098848728967&{}:{}&0.190033807112&{}:{}&0.711117463921&,\\ A^*_{\mathbf{1}}&{}\approx{}&0.000000000000&{}:{}&-0.124844448887&{}:{}&1.124844448887&,\\B^*_{\mathbf{1}}&{}\approx{}&-0.057677535825&{}:{}&0.000000000000&{}:{}&1.057677535825&,\\C^*_{\mathbf{1}}&{}\approx{}&0.329459311299&{}:{}&0.670540688701&{}:{}&0.000000000000&. \end{alignedat} \]
1 (002)

Hiroyasu Kamo