Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


[Top] > Pythagorean Triangle U > 7c (332)

7c(332)

Malfatti circles

7c (332)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&, \\ P_{\mathbf{7c}}&{}\approx{}&-0.024425610155&{}:{}&-0.034698448318&{}:{}&1.059124058474&, \\ P^-_{\mathbf{7c}}&{}\approx{}&0.007462751586&{}:{}&0.041235024083&{}:{}&0.951302224331&, \\ P^+_{\mathbf{7c}}&{}\approx{}&-0.057993843948&{}:{}&-0.114632079747&{}:{}&1.172625923695&, \\ Q_{\mathbf{7c}}&{}\approx{}&0.217500699509&{}:{}&0.141988640760&{}:{}&0.640510659731&, \\ I^\prime_{\mathbf{7c}}&{}\approx{}&-0.128676649241&{}:{}&-0.237595209797&{}:{}&1.366271859037&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{7c}}\)
\(P^-_{\mathbf{7c}}\)
\(P^+_{\mathbf{7c}}\)
\(Q_{\mathbf{7c}}\)
\(I^\prime_{\mathbf{7c}}\)
7c (332)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7c}}&{}\approx{}&0.544004113661&{}:{}&-5.471950636069&{}:{}&5.927946522408&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.342656599941&{}:{}&0.451749440095&{}:{}&0.890907159846&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.037016188229&{}:{}&-0.088838851750&{}:{}&1.125855039979&, \\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.066814591915&{}:{}&-0.036134210836&{}:{}&1.102948802750&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.027733841437&{}:{}&-0.174839134769&{}:{}&1.202572976206&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.226811266559&{}:{}&-0.322202760166&{}:{}&1.549014026725&, \\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.032537698732&{}:{}&0.042896845375&{}:{}&0.989640853358&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.008444895276&{}:{}&-0.084944302654&{}:{}&1.076499407379&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.057397626497&{}:{}&0.317147433291&{}:{}&0.625454940213&, \\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.181455294172&{}:{}&0.818544705828&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.253493962716&{}:{}&0.000000000000&{}:{}&0.746506037284&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.605026839867&{}:{}&0.394973160133&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7c}}}{B^\prime_{\mathbf{7c}}}{C^\prime_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^*_{\mathbf{7c}}}{B^*_{\mathbf{7c}}}{C^*_{\mathbf{7c}}}\)
7c (332)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7c}}}}&{}\approx{}&-1.823983545356&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.274125279953&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.029612950583&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.544004113661&{}:{}&-5.471950636069&{}:{}&5.927946522408&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.342656599941&{}:{}&0.451749440095&{}:{}&0.890907159846&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.037016188229&{}:{}&-0.088838851750&{}:{}&1.125855039979&. \end{alignedat} \]
7c (332)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7c}}&{}\approx{}&-0.024425610155&{}:{}&-0.034698448318&{}:{}&1.059124058474&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.066814591915&{}:{}&-0.036134210836&{}:{}&1.102948802750&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.027733841437&{}:{}&-0.174839134769&{}:{}&1.202572976206&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.226811266559&{}:{}&-0.322202760166&{}:{}&1.549014026725&. \end{alignedat} \]
7c (332)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7c}}&{}\approx{}&0.007462751586&{}:{}&0.041235024083&{}:{}&0.951302224331&,\\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.032537698732&{}:{}&0.042896845375&{}:{}&0.989640853358&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.008444895276&{}:{}&-0.084944302654&{}:{}&1.076499407379&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.057397626497&{}:{}&0.317147433291&{}:{}&0.625454940213&. \end{alignedat} \]
7c (332)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7c}}&{}\approx{}&-0.057993843948&{}:{}&-0.114632079747&{}:{}&1.172625923695&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.544004113661&{}:{}&-5.471950636069&{}:{}&5.927946522408&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.342656599941&{}:{}&0.451749440095&{}:{}&0.890907159846&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.037016188229&{}:{}&-0.088838851750&{}:{}&1.125855039979&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.066814591915&{}:{}&-0.036134210836&{}:{}&1.102948802750&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.027733841437&{}:{}&-0.174839134769&{}:{}&1.202572976206&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.226811266559&{}:{}&-0.322202760166&{}:{}&1.549014026725&, \end{alignedat} \]
7c (332)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7c}}&{}\approx{}&0.217500699509&{}:{}&0.141988640760&{}:{}&0.640510659731&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.181455294172&{}:{}&0.818544705828&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.253493962716&{}:{}&0.000000000000&{}:{}&0.746506037284&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.605026839867&{}:{}&0.394973160133&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7c}}&{}\approx{}&-0.128676649241&{}:{}&-0.237595209797&{}:{}&1.366271859037&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.066814591915&{}:{}&-0.036134210836&{}:{}&1.102948802750&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.027733841437&{}:{}&-0.174839134769&{}:{}&1.202572976206&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.226811266559&{}:{}&-0.322202760166&{}:{}&1.549014026725&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.181455294172&{}:{}&0.818544705828&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.253493962716&{}:{}&0.000000000000&{}:{}&0.746506037284&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.605026839867&{}:{}&0.394973160133&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)