Derousseau's Generalization of the Malfatti circles

Equilateral Triangle

\(a:b:c=1:1:1\), \(A=60\degree\), \(B=60\degree\), \(C=60\degree\).


[Top] > Equilateral Triangle > 5 (202)

5(202)

Malfatti circles

5 (202)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&, \\ P_{\mathbf{5}}&{}\approx{}&0.498714618890&{}:{}&0.002570762220&{}:{}&0.498714618890&, \\ P^-_{\mathbf{5}}&{}\approx{}&0.609378523886&{}:{}&-0.218757047773&{}:{}&0.609378523886&, \\ P^+_{\mathbf{5}}&{}\approx{}&0.451387728771&{}:{}&0.097224542457&{}:{}&0.451387728771&, \\ Q_{\mathbf{5}}&{}\approx{}&0.518617531751&{}:{}&-0.037235063502&{}:{}&0.518617531751&, \\ I^\prime_{\mathbf{5}}&{}\approx{}&0.482672825160&{}:{}&0.034654349680&{}:{}&0.482672825160&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{5}}\)
\(P^-_{\mathbf{5}}\)
\(P^+_{\mathbf{5}}\)
\(Q_{\mathbf{5}}\)
\(I^\prime_{\mathbf{5}}\)
5 (202)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5}}&{}\approx{}&0.577350269190&{}:{}&0.211324865405&{}:{}&0.211324865405&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.981125224325&{}:{}&-0.962250448649&{}:{}&0.981125224325&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.211324865405&{}:{}&0.211324865405&{}:{}&0.577350269190&, \\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.347748149397&{}:{}&0.003344969709&{}:{}&0.648906880894&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.394337567297&{}:{}&0.211324865405&{}:{}&0.394337567297&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.648906880894&{}:{}&0.003344969709&{}:{}&0.347748149397&, \\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.363466037631&{}:{}&-0.356473719264&{}:{}&0.993007681634&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.422649730810&{}:{}&0.154700538379&{}:{}&0.422649730810&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.993007681634&{}:{}&-0.356473719264&{}:{}&0.363466037631&, \\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.077350269190&{}:{}&1.077350269190&,\\B^*_{\mathbf{5}}&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*_{\mathbf{5}}&{}\approx{}&1.077350269190&{}:{}&-0.077350269190&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5}}}{B^\prime_{\mathbf{5}}}{C^\prime_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5}}}{B^{\prime\prime}_{\mathbf{5}}}{C^{\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5}}}{B^{\prime\prime\prime}_{\mathbf{5}}}{C^{\prime\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^*_{\mathbf{5}}}{B^*_{\mathbf{5}}}{C^*_{\mathbf{5}}}\)
5 (202)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5}}}}&{}\approx{}&0.633974596216&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5}}}}&{}\approx{}&2.943375672974&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5}}}}&{}\approx{}&0.633974596216&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.333333333333&{}:{}&0.333333333333&{}:{}&0.333333333333&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.577350269190&{}:{}&0.211324865405&{}:{}&0.211324865405&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.981125224325&{}:{}&-0.962250448649&{}:{}&0.981125224325&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.211324865405&{}:{}&0.211324865405&{}:{}&0.577350269190&. \end{alignedat} \]
5 (202)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5}}&{}\approx{}&0.498714618890&{}:{}&0.002570762220&{}:{}&0.498714618890&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.347748149397&{}:{}&0.003344969709&{}:{}&0.648906880894&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.394337567297&{}:{}&0.211324865405&{}:{}&0.394337567297&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.648906880894&{}:{}&0.003344969709&{}:{}&0.347748149397&. \end{alignedat} \]
5 (202)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5}}&{}\approx{}&0.609378523886&{}:{}&-0.218757047773&{}:{}&0.609378523886&,\\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.363466037631&{}:{}&-0.356473719264&{}:{}&0.993007681634&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.422649730810&{}:{}&0.154700538379&{}:{}&0.422649730810&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.993007681634&{}:{}&-0.356473719264&{}:{}&0.363466037631&. \end{alignedat} \]
5 (202)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5}}&{}\approx{}&0.451387728771&{}:{}&0.097224542457&{}:{}&0.451387728771&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.577350269190&{}:{}&0.211324865405&{}:{}&0.211324865405&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.981125224325&{}:{}&-0.962250448649&{}:{}&0.981125224325&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.211324865405&{}:{}&0.211324865405&{}:{}&0.577350269190&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.347748149397&{}:{}&0.003344969709&{}:{}&0.648906880894&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.394337567297&{}:{}&0.211324865405&{}:{}&0.394337567297&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.648906880894&{}:{}&0.003344969709&{}:{}&0.347748149397&, \end{alignedat} \]
5 (202)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5}}&{}\approx{}&0.518617531751&{}:{}&-0.037235063502&{}:{}&0.518617531751&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.077350269190&{}:{}&1.077350269190&,\\B^*_{\mathbf{5}}&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*_{\mathbf{5}}&{}\approx{}&1.077350269190&{}:{}&-0.077350269190&{}:{}&0.000000000000&. \end{alignedat} \]
5 (202)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5}}&{}\approx{}&0.482672825160&{}:{}&0.034654349680&{}:{}&0.482672825160&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.347748149397&{}:{}&0.003344969709&{}:{}&0.648906880894&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.394337567297&{}:{}&0.211324865405&{}:{}&0.394337567297&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.648906880894&{}:{}&0.003344969709&{}:{}&0.347748149397&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.077350269190&{}:{}&1.077350269190&,\\B^*_{\mathbf{5}}&{}\approx{}&0.500000000000&{}:{}&0.000000000000&{}:{}&0.500000000000&,\\C^*_{\mathbf{5}}&{}\approx{}&1.077350269190&{}:{}&-0.077350269190&{}:{}&0.000000000000&. \end{alignedat} \]
5 (202)