Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 108° Top Angle

\(A=36\degree\), \(B=36\degree\), \(C=108\degree\).

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[Top] > Isosceles Triangle with 108° Top Angle > 1c (112)

1c(112)

Malfatti circles

1c (112)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&, \\ P_{\mathbf{1c}}&{}\approx{}&-0.392698881462&{}:{}&-0.392698881462&{}:{}&1.785397762923&, \\ P^-_{\mathbf{1c}}&{}\approx{}&0.070755769515&{}:{}&0.070755769515&{}:{}&0.858488460969&, \\ P^+_{\mathbf{1c}}&{}\approx{}&-1.062303627662&{}:{}&-1.062303627662&{}:{}&3.124607255324&, \\ Q_{\mathbf{1c}}&{}\approx{}&1.741917497203&{}:{}&1.741917497203&{}:{}&-2.483834994406&, \\ I^\prime_{\mathbf{1c}}&{}\approx{}&-1.403979697400&{}:{}&-1.403979697400&{}:{}&3.807959394800&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{1c}}\)
\(P^-_{\mathbf{1c}}\)
\(P^+_{\mathbf{1c}}\)
\(Q_{\mathbf{1c}}\)
\(I^\prime_{\mathbf{1c}}\)
1c (112)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1c}}&{}\approx{}&0.221231742082&{}:{}&-1.260073510670&{}:{}&2.038841768588&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.260073510670&{}:{}&0.221231742082&{}:{}&2.038841768588&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-7.679397834345&{}:{}&-7.679397834345&{}:{}&16.358795668690&, \\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-2.164916786573&{}:{}&-0.892410626985&{}:{}&4.057327413558&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.892410626985&{}:{}&-2.164916786573&{}:{}&4.057327413558&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.519420884294&{}:{}&-0.519420884294&{}:{}&2.038841768588&, \\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.765820483455&{}:{}&0.134455488702&{}:{}&1.631364994753&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.134455488702&{}:{}&-0.765820483455&{}:{}&1.631364994753&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.089159421976&{}:{}&0.089159421976&{}:{}&0.821681156047&, \\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-2.347858762963&{}:{}&3.347858762963&,\\B^*_{\mathbf{1c}}&{}\approx{}&-2.347858762963&{}:{}&0.000000000000&{}:{}&3.347858762963&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1c}}}{B^\prime_{\mathbf{1c}}}{C^\prime_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^*_{\mathbf{1c}}}{B^*_{\mathbf{1c}}}{C^*_{\mathbf{1c}}}\)
1c (112)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.481305252753&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.481305252753&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1c}}}}&{}\approx{}&-2.933268959588&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.221231742082&{}:{}&-1.260073510670&{}:{}&2.038841768588&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.260073510670&{}:{}&0.221231742082&{}:{}&2.038841768588&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-7.679397834345&{}:{}&-7.679397834345&{}:{}&16.358795668690&. \end{alignedat} \]
1c (112)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1c}}&{}\approx{}&-0.392698881462&{}:{}&-0.392698881462&{}:{}&1.785397762923&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-2.164916786573&{}:{}&-0.892410626985&{}:{}&4.057327413558&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.892410626985&{}:{}&-2.164916786573&{}:{}&4.057327413558&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.519420884294&{}:{}&-0.519420884294&{}:{}&2.038841768588&. \end{alignedat} \]
1c (112)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1c}}&{}\approx{}&0.070755769515&{}:{}&0.070755769515&{}:{}&0.858488460969&,\\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.765820483455&{}:{}&0.134455488702&{}:{}&1.631364994753&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.134455488702&{}:{}&-0.765820483455&{}:{}&1.631364994753&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.089159421976&{}:{}&0.089159421976&{}:{}&0.821681156047&. \end{alignedat} \]
1c (112)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1c}}&{}\approx{}&-1.062303627662&{}:{}&-1.062303627662&{}:{}&3.124607255324&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.221231742082&{}:{}&-1.260073510670&{}:{}&2.038841768588&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.260073510670&{}:{}&0.221231742082&{}:{}&2.038841768588&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-7.679397834345&{}:{}&-7.679397834345&{}:{}&16.358795668690&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-2.164916786573&{}:{}&-0.892410626985&{}:{}&4.057327413558&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.892410626985&{}:{}&-2.164916786573&{}:{}&4.057327413558&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.519420884294&{}:{}&-0.519420884294&{}:{}&2.038841768588&, \end{alignedat} \]
1c (112)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1c}}&{}\approx{}&1.741917497203&{}:{}&1.741917497203&{}:{}&-2.483834994406&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-2.347858762963&{}:{}&3.347858762963&,\\B^*_{\mathbf{1c}}&{}\approx{}&-2.347858762963&{}:{}&0.000000000000&{}:{}&3.347858762963&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1c}}&{}\approx{}&-1.403979697400&{}:{}&-1.403979697400&{}:{}&3.807959394800&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-2.164916786573&{}:{}&-0.892410626985&{}:{}&4.057327413558&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.892410626985&{}:{}&-2.164916786573&{}:{}&4.057327413558&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.519420884294&{}:{}&-0.519420884294&{}:{}&2.038841768588&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-2.347858762963&{}:{}&3.347858762963&,\\B^*_{\mathbf{1c}}&{}\approx{}&-2.347858762963&{}:{}&0.000000000000&{}:{}&3.347858762963&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)