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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7(222)

Malfatti circles

7 (222)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&, \\ P_{\mathbf{7}}&{}\approx{}&0.494361738666&{}:{}&0.494361738666&{}:{}&0.011276522668&, \\ P^-_{\mathbf{7}}&{}\approx{}&0.090019096795&{}:{}&0.090019096795&{}:{}&0.819961806411&, \\ P^+_{\mathbf{7}}&{}\approx{}&0.345163372007&{}:{}&0.345163372007&{}:{}&0.309673255986&, \\ Q_{\mathbf{7}}&{}\approx{}&0.432741767197&{}:{}&0.432741767197&{}:{}&0.134516465606&, \\ I^\prime_{\mathbf{7}}&{}\approx{}&0.456180656796&{}:{}&0.456180656796&{}:{}&0.087638686407&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{7}}\)
\(P^-_{\mathbf{7}}\)
\(P^+_{\mathbf{7}}\)
\(Q_{\mathbf{7}}\)
\(I^\prime_{\mathbf{7}}\)
7 (222)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7}}&{}\approx{}&-0.071882550435&{}:{}&0.409422702318&{}:{}&0.662459848116&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.409422702318&{}:{}&-0.071882550435&{}:{}&0.662459848116&,\\C^\prime_{\mathbf{7}}&{}\approx{}&2.495187611587&{}:{}&2.495187611587&{}:{}&-3.990375223174&, \\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.703424104663&{}:{}&0.289961789835&{}:{}&0.006614105502&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.289961789835&{}:{}&0.703424104663&{}:{}&0.006614105502&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.168770075942&{}:{}&0.168770075942&{}:{}&0.662459848116&, \\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-1.290657256567&{}:{}&0.226601345782&{}:{}&2.064055910785&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.226601345782&{}:{}&-1.290657256567&{}:{}&2.064055910785&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.138583532638&{}:{}&-0.138583532638&{}:{}&1.277167065277&, \\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.762865556060&{}:{}&0.237134443940&,\\B^*_{\mathbf{7}}&{}\approx{}&0.762865556060&{}:{}&0.000000000000&{}:{}&0.237134443940&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7}}}{B^\prime_{\mathbf{7}}}{C^\prime_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7}}}{B^{\prime\prime}_{\mathbf{7}}}{C^{\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7}}}{B^{\prime\prime\prime}_{\mathbf{7}}}{C^{\prime\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^*_{\mathbf{7}}}{B^*_{\mathbf{7}}}{C^*_{\mathbf{7}}}\)
7 (222)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7}}}}&{}\approx{}&1.481305252753&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7}}}}&{}\approx{}&1.481305252753&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7}}}}&{}\approx{}&9.027673587030&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.276393202250&{}:{}&0.276393202250&{}:{}&0.447213595500&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&-0.071882550435&{}:{}&0.409422702318&{}:{}&0.662459848116&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.409422702318&{}:{}&-0.071882550435&{}:{}&0.662459848116&,\\C^\prime_{\mathbf{7}}&{}\approx{}&2.495187611587&{}:{}&2.495187611587&{}:{}&-3.990375223174&. \end{alignedat} \]
7 (222)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7}}&{}\approx{}&0.494361738666&{}:{}&0.494361738666&{}:{}&0.011276522668&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.703424104663&{}:{}&0.289961789835&{}:{}&0.006614105502&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.289961789835&{}:{}&0.703424104663&{}:{}&0.006614105502&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.168770075942&{}:{}&0.168770075942&{}:{}&0.662459848116&. \end{alignedat} \]
7 (222)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7}}&{}\approx{}&0.090019096795&{}:{}&0.090019096795&{}:{}&0.819961806411&,\\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-1.290657256567&{}:{}&0.226601345782&{}:{}&2.064055910785&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.226601345782&{}:{}&-1.290657256567&{}:{}&2.064055910785&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.138583532638&{}:{}&-0.138583532638&{}:{}&1.277167065277&. \end{alignedat} \]
7 (222)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7}}&{}\approx{}&0.345163372007&{}:{}&0.345163372007&{}:{}&0.309673255986&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&-0.071882550435&{}:{}&0.409422702318&{}:{}&0.662459848116&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.409422702318&{}:{}&-0.071882550435&{}:{}&0.662459848116&,\\C^\prime_{\mathbf{7}}&{}\approx{}&2.495187611587&{}:{}&2.495187611587&{}:{}&-3.990375223174&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.703424104663&{}:{}&0.289961789835&{}:{}&0.006614105502&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.289961789835&{}:{}&0.703424104663&{}:{}&0.006614105502&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.168770075942&{}:{}&0.168770075942&{}:{}&0.662459848116&, \end{alignedat} \]
7 (222)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7}}&{}\approx{}&0.432741767197&{}:{}&0.432741767197&{}:{}&0.134516465606&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.762865556060&{}:{}&0.237134443940&,\\B^*_{\mathbf{7}}&{}\approx{}&0.762865556060&{}:{}&0.000000000000&{}:{}&0.237134443940&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7}}&{}\approx{}&0.456180656796&{}:{}&0.456180656796&{}:{}&0.087638686407&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.703424104663&{}:{}&0.289961789835&{}:{}&0.006614105502&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.289961789835&{}:{}&0.703424104663&{}:{}&0.006614105502&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.168770075942&{}:{}&0.168770075942&{}:{}&0.662459848116&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.762865556060&{}:{}&0.237134443940&,\\B^*_{\mathbf{7}}&{}\approx{}&0.762865556060&{}:{}&0.000000000000&{}:{}&0.237134443940&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)