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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1a (013)

Malfatti circles

1a (013)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&, \\ P_{\mathbf{1a}}&{}\approx{}&-0.035175362399&{}:{}&0.941875402474&{}:{}&0.093299959924&, \\ P^-_{\mathbf{1a}}&{}\approx{}&0.005996690905&{}:{}&0.980191578450&{}:{}&0.013811730645&, \\ P^+_{\mathbf{1a}}&{}\approx{}&-0.068142372231&{}:{}&0.911195131543&{}:{}&0.156947240687&, \\ Q_{\mathbf{1a}}&{}\approx{}&0.090826788008&{}:{}&1.194938298938&{}:{}&-0.285765086946&, \\ I^\prime_{\mathbf{1a}}&{}\approx{}&-0.123253008728&{}:{}&0.839373551747&{}:{}&0.283879456981&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{1a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{1a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{1a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{1a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{1a}}\) Radical center of the Malfatti circles
1a (013)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1a}}&{}\approx{}&0.034074173711&{}:{}&0.448287736084&{}:{}&0.517638090205&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.044406327739&{}:{}&0.955593672261&{}:{}&0.088812655477&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.965925826289&{}:{}&1.673032607476&{}:{}&0.292893218813&, \\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.084909143280&{}:{}&0.987126696686&{}:{}&0.097782446595&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.177250691694&{}:{}&0.707106781187&{}:{}&0.470143910508&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.032913124397&{}:{}&0.881300437976&{}:{}&0.151612686421&, \\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.048024884147&{}:{}&1.033462520746&{}:{}&0.014562363401&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.060755344153&{}:{}&0.799311405657&{}:{}&0.139933250190&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.005566474518&{}:{}&0.909870382050&{}:{}&0.084563143432&, \\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.314313139868&{}:{}&-0.314313139868&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.465925826289&{}:{}&0.000000000000&{}:{}&1.465925826289&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.070640266196&{}:{}&0.929359733804&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1a}}}{B^\prime_{\mathbf{1a}}}{C^\prime_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^*_{\mathbf{1a}}}{B^*_{\mathbf{1a}}}{C^*_{\mathbf{1a}}}\)
1a (013)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1a}}}}&{}\approx{}&0.707106781187&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1a}}}}&{}\approx{}&0.121320343560&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1a}}}}&{}\approx{}&2.638958433765&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&0.034074173711&{}:{}&0.448287736084&{}:{}&0.517638090205&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.044406327739&{}:{}&0.955593672261&{}:{}&0.088812655477&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.965925826289&{}:{}&1.673032607476&{}:{}&0.292893218813&. \end{alignedat} \]
1a (013)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1a}}&{}\approx{}&-0.035175362399&{}:{}&0.941875402474&{}:{}&0.093299959924&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.084909143280&{}:{}&0.987126696686&{}:{}&0.097782446595&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.177250691694&{}:{}&0.707106781187&{}:{}&0.470143910508&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.032913124397&{}:{}&0.881300437976&{}:{}&0.151612686421&. \end{alignedat} \]
1a (013)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1a}}&{}\approx{}&0.005996690905&{}:{}&0.980191578450&{}:{}&0.013811730645&,\\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.048024884147&{}:{}&1.033462520746&{}:{}&0.014562363401&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.060755344153&{}:{}&0.799311405657&{}:{}&0.139933250190&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.005566474518&{}:{}&0.909870382050&{}:{}&0.084563143432&. \end{alignedat} \]
1a (013)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1a}}&{}\approx{}&-0.068142372231&{}:{}&0.911195131543&{}:{}&0.156947240687&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&0.034074173711&{}:{}&0.448287736084&{}:{}&0.517638090205&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.044406327739&{}:{}&0.955593672261&{}:{}&0.088812655477&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.965925826289&{}:{}&1.673032607476&{}:{}&0.292893218813&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.084909143280&{}:{}&0.987126696686&{}:{}&0.097782446595&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.177250691694&{}:{}&0.707106781187&{}:{}&0.470143910508&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.032913124397&{}:{}&0.881300437976&{}:{}&0.151612686421&, \end{alignedat} \]
1a (013)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1a}}&{}\approx{}&0.090826788008&{}:{}&1.194938298938&{}:{}&-0.285765086946&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.314313139868&{}:{}&-0.314313139868&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.465925826289&{}:{}&0.000000000000&{}:{}&1.465925826289&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.070640266196&{}:{}&0.929359733804&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1a}}&{}\approx{}&-0.123253008728&{}:{}&0.839373551747&{}:{}&0.283879456981&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.084909143280&{}:{}&0.987126696686&{}:{}&0.097782446595&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.177250691694&{}:{}&0.707106781187&{}:{}&0.470143910508&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.032913124397&{}:{}&0.881300437976&{}:{}&0.151612686421&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.314313139868&{}:{}&-0.314313139868&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.465925826289&{}:{}&0.000000000000&{}:{}&1.465925826289&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.070640266196&{}:{}&0.929359733804&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)