Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 135° Top Angle

\(A=22.5\degree\), \(B=22.5\degree\), \(C=135\degree\).


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7(222)

Malfatti circles

7 (222)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&, \\ P_{\mathbf{7}}&{}\approx{}&0.499552854332&{}:{}&0.499552854332&{}:{}&0.000894291336&, \\ P^-_{\mathbf{7}}&{}\approx{}&-0.170483256580&{}:{}&-0.170483256580&{}:{}&1.340966513159&, \\ P^+_{\mathbf{7}}&{}\approx{}&0.353623965370&{}:{}&0.353623965370&{}:{}&0.292752069260&, \\ Q_{\mathbf{7}}&{}\approx{}&0.465678818685&{}:{}&0.465678818685&{}:{}&0.068642362630&, \\ I^\prime_{\mathbf{7}}&{}\approx{}&0.486023490490&{}:{}&0.486023490490&{}:{}&0.027953019020&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{7}}\)
\(P^-_{\mathbf{7}}\)
\(P^+_{\mathbf{7}}\)
\(Q_{\mathbf{7}}\)
\(I^\prime_{\mathbf{7}}\)
7 (222)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7}}&{}\approx{}&0.076120467489&{}:{}&0.324423348821&{}:{}&0.599456183690&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.324423348821&{}:{}&0.076120467489&{}:{}&0.599456183690&,\\C^\prime_{\mathbf{7}}&{}\approx{}&6.474538827970&{}:{}&6.474538827970&{}:{}&-11.949077655939&, \\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.617885940244&{}:{}&0.381431227821&{}:{}&0.000682831935&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.381431227821&{}:{}&0.617885940244&{}:{}&0.000682831935&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.200271908155&{}:{}&0.200271908155&{}:{}&0.599456183690&, \\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-1.636887027219&{}:{}&-0.384067935279&{}:{}&3.020954962498&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.384067935279&{}:{}&-1.636887027219&{}:{}&3.020954962498&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.101264812114&{}:{}&0.101264812114&{}:{}&0.797470375771&, \\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.871533517609&{}:{}&0.128466482391&,\\B^*_{\mathbf{7}}&{}\approx{}&0.871533517609&{}:{}&0.000000000000&{}:{}&0.128466482391&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7}}}{B^\prime_{\mathbf{7}}}{C^\prime_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7}}}{B^{\prime\prime}_{\mathbf{7}}}{C^{\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7}}}{B^{\prime\prime\prime}_{\mathbf{7}}}{C^{\prime\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^*_{\mathbf{7}}}{B^*_{\mathbf{7}}}{C^*_{\mathbf{7}}}\)
7 (222)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7}}}}&{}\approx{}&1.248302881333&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7}}}}&{}\approx{}&1.248302881333&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7}}}}&{}\approx{}&24.912465467161&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.259891532474&{}:{}&0.259891532474&{}:{}&0.480216935052&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.076120467489&{}:{}&0.324423348821&{}:{}&0.599456183690&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.324423348821&{}:{}&0.076120467489&{}:{}&0.599456183690&,\\C^\prime_{\mathbf{7}}&{}\approx{}&6.474538827970&{}:{}&6.474538827970&{}:{}&-11.949077655939&. \end{alignedat} \]
7 (222)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7}}&{}\approx{}&0.499552854332&{}:{}&0.499552854332&{}:{}&0.000894291336&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.617885940244&{}:{}&0.381431227821&{}:{}&0.000682831935&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.381431227821&{}:{}&0.617885940244&{}:{}&0.000682831935&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.200271908155&{}:{}&0.200271908155&{}:{}&0.599456183690&. \end{alignedat} \]
7 (222)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7}}&{}\approx{}&-0.170483256580&{}:{}&-0.170483256580&{}:{}&1.340966513159&,\\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-1.636887027219&{}:{}&-0.384067935279&{}:{}&3.020954962498&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.384067935279&{}:{}&-1.636887027219&{}:{}&3.020954962498&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.101264812114&{}:{}&0.101264812114&{}:{}&0.797470375771&. \end{alignedat} \]
7 (222)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7}}&{}\approx{}&0.353623965370&{}:{}&0.353623965370&{}:{}&0.292752069260&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.076120467489&{}:{}&0.324423348821&{}:{}&0.599456183690&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.324423348821&{}:{}&0.076120467489&{}:{}&0.599456183690&,\\C^\prime_{\mathbf{7}}&{}\approx{}&6.474538827970&{}:{}&6.474538827970&{}:{}&-11.949077655939&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.617885940244&{}:{}&0.381431227821&{}:{}&0.000682831935&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.381431227821&{}:{}&0.617885940244&{}:{}&0.000682831935&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.200271908155&{}:{}&0.200271908155&{}:{}&0.599456183690&, \end{alignedat} \]
7 (222)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7}}&{}\approx{}&0.465678818685&{}:{}&0.465678818685&{}:{}&0.068642362630&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.871533517609&{}:{}&0.128466482391&,\\B^*_{\mathbf{7}}&{}\approx{}&0.871533517609&{}:{}&0.000000000000&{}:{}&0.128466482391&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7}}&{}\approx{}&0.486023490490&{}:{}&0.486023490490&{}:{}&0.027953019020&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.617885940244&{}:{}&0.381431227821&{}:{}&0.000682831935&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.381431227821&{}:{}&0.617885940244&{}:{}&0.000682831935&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.200271908155&{}:{}&0.200271908155&{}:{}&0.599456183690&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.871533517609&{}:{}&0.128466482391&,\\B^*_{\mathbf{7}}&{}\approx{}&0.871533517609&{}:{}&0.000000000000&{}:{}&0.128466482391&,\\C^*_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)