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 > 7c (332)

7c(332)

Malfatti circles

7c (332)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&, \\ P_{\mathbf{7c}}&{}\approx{}&-0.069848712968&{}:{}&-0.069848712968&{}:{}&1.139697425937&, \\ P^-_{\mathbf{7c}}&{}\approx{}&0.014408693869&{}:{}&0.014408693869&{}:{}&0.971182612263&, \\ P^+_{\mathbf{7c}}&{}\approx{}&-0.159741906973&{}:{}&-0.159741906973&{}:{}&1.319483813946&, \\ Q_{\mathbf{7c}}&{}\approx{}&0.212707050415&{}:{}&0.212707050415&{}:{}&0.574585899170&, \\ I^\prime_{\mathbf{7c}}&{}\approx{}&-0.318655933735&{}:{}&-0.318655933735&{}:{}&1.637311867470&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{7c}}\)
\(P^-_{\mathbf{7c}}\)
\(P^+_{\mathbf{7c}}\)
\(Q_{\mathbf{7c}}\)
\(I^\prime_{\mathbf{7c}}\)
7c (332)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7c}}&{}\approx{}&0.221231742082&{}:{}&-1.260073510670&{}:{}&2.038841768588&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.260073510670&{}:{}&0.221231742082&{}:{}&2.038841768588&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.101689070346&{}:{}&-0.101689070346&{}:{}&1.203378140691&, \\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.188190960236&{}:{}&-0.077575089194&{}:{}&1.265766049430&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.077575089194&{}:{}&-0.188190960236&{}:{}&1.265766049430&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.519420884294&{}:{}&-0.519420884294&{}:{}&2.038841768588&, \\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.090830901682&{}:{}&0.015947227240&{}:{}&1.074883674442&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.015947227240&{}:{}&-0.090830901682&{}:{}&1.074883674442&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.089159421976&{}:{}&0.089159421976&{}:{}&0.821681156047&, \\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.270175225787&{}:{}&0.729824774213&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.270175225787&{}:{}&0.000000000000&{}:{}&0.729824774213&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7c}}}{B^\prime_{\mathbf{7c}}}{C^\prime_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^*_{\mathbf{7c}}}{B^*_{\mathbf{7c}}}{C^*_{\mathbf{7c}}}\)
7c (332)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.481305252753&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.481305252753&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.038841768588&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.221231742082&{}:{}&-1.260073510670&{}:{}&2.038841768588&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.260073510670&{}:{}&0.221231742082&{}:{}&2.038841768588&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.101689070346&{}:{}&-0.101689070346&{}:{}&1.203378140691&. \end{alignedat} \]
7c (332)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7c}}&{}\approx{}&-0.069848712968&{}:{}&-0.069848712968&{}:{}&1.139697425937&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.188190960236&{}:{}&-0.077575089194&{}:{}&1.265766049430&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.077575089194&{}:{}&-0.188190960236&{}:{}&1.265766049430&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.519420884294&{}:{}&-0.519420884294&{}:{}&2.038841768588&. \end{alignedat} \]
7c (332)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7c}}&{}\approx{}&0.014408693869&{}:{}&0.014408693869&{}:{}&0.971182612263&,\\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.090830901682&{}:{}&0.015947227240&{}:{}&1.074883674442&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.015947227240&{}:{}&-0.090830901682&{}:{}&1.074883674442&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.089159421976&{}:{}&0.089159421976&{}:{}&0.821681156047&. \end{alignedat} \]
7c (332)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7c}}&{}\approx{}&-0.159741906973&{}:{}&-0.159741906973&{}:{}&1.319483813946&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.221231742082&{}:{}&-1.260073510670&{}:{}&2.038841768588&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.260073510670&{}:{}&0.221231742082&{}:{}&2.038841768588&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.101689070346&{}:{}&-0.101689070346&{}:{}&1.203378140691&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.188190960236&{}:{}&-0.077575089194&{}:{}&1.265766049430&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.077575089194&{}:{}&-0.188190960236&{}:{}&1.265766049430&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.519420884294&{}:{}&-0.519420884294&{}:{}&2.038841768588&, \end{alignedat} \]
7c (332)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7c}}&{}\approx{}&0.212707050415&{}:{}&0.212707050415&{}:{}&0.574585899170&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.270175225787&{}:{}&0.729824774213&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.270175225787&{}:{}&0.000000000000&{}:{}&0.729824774213&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7c}}&{}\approx{}&-0.318655933735&{}:{}&-0.318655933735&{}:{}&1.637311867470&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.188190960236&{}:{}&-0.077575089194&{}:{}&1.265766049430&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.077575089194&{}:{}&-0.188190960236&{}:{}&1.265766049430&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.519420884294&{}:{}&-0.519420884294&{}:{}&2.038841768588&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.270175225787&{}:{}&0.729824774213&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.270175225787&{}:{}&0.000000000000&{}:{}&0.729824774213&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)