Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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5c(312)

Malfatti circles

5c (312)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&, \\ P_{\mathbf{5c}}&{}\approx{}&-0.055728494635&{}:{}&-0.035758712244&{}:{}&1.091487206880&, \\ P^-_{\mathbf{5c}}&{}\approx{}&-0.006290075653&{}:{}&0.079183338695&{}:{}&0.927106736958&, \\ P^+_{\mathbf{5c}}&{}\approx{}&-0.109217382054&{}:{}&-0.160117916057&{}:{}&1.269335298110&, \\ Q_{\mathbf{5c}}&{}\approx{}&-0.705254708953&{}:{}&0.309427494852&{}:{}&1.395827214101&, \\ I^\prime_{\mathbf{5c}}&{}\approx{}&-0.204286721104&{}:{}&-0.253511717770&{}:{}&1.457798438875&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{5c}}\)
\(P^-_{\mathbf{5c}}\)
\(P^+_{\mathbf{5c}}\)
\(Q_{\mathbf{5c}}\)
\(I^\prime_{\mathbf{5c}}\)
5c (312)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5c}}&{}\approx{}&-1.980750764737&{}:{}&-35.769009176847&{}:{}&38.749759941585&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.305524282344&{}:{}&0.511161148250&{}:{}&0.794363134093&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.061771066627&{}:{}&-0.148250559905&{}:{}&1.210021626533&, \\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.102765041016&{}:{}&-0.037351893006&{}:{}&1.140116934021&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.069691765945&{}:{}&-0.295277294348&{}:{}&1.364969060293&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.339176265378&{}:{}&-0.217635637804&{}:{}&1.556811903183&, \\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.049353783090&{}:{}&0.082571952191&{}:{}&0.966781830899&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.007792186975&{}:{}&-0.140713719448&{}:{}&1.148505906423&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.032100891654&{}:{}&0.404105755843&{}:{}&0.627995135811&, \\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.181455294172&{}:{}&0.818544705828&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.021260915682&{}:{}&0.000000000000&{}:{}&2.021260915682&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.781723650696&{}:{}&-0.781723650696&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5c}}}{B^\prime_{\mathbf{5c}}}{C^\prime_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^*_{\mathbf{5c}}}{B^*_{\mathbf{5c}}}{C^*_{\mathbf{5c}}}\)
5c (312)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5c}}}}&{}\approx{}&-11.923003058949&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.244419425875&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.049416853302&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-1.980750764737&{}:{}&-35.769009176847&{}:{}&38.749759941585&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.305524282344&{}:{}&0.511161148250&{}:{}&0.794363134093&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.061771066627&{}:{}&-0.148250559905&{}:{}&1.210021626533&. \end{alignedat} \]
5c (312)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5c}}&{}\approx{}&-0.055728494635&{}:{}&-0.035758712244&{}:{}&1.091487206880&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.102765041016&{}:{}&-0.037351893006&{}:{}&1.140116934021&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.069691765945&{}:{}&-0.295277294348&{}:{}&1.364969060293&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.339176265378&{}:{}&-0.217635637804&{}:{}&1.556811903183&. \end{alignedat} \]
5c (312)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5c}}&{}\approx{}&-0.006290075653&{}:{}&0.079183338695&{}:{}&0.927106736958&,\\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.049353783090&{}:{}&0.082571952191&{}:{}&0.966781830899&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.007792186975&{}:{}&-0.140713719448&{}:{}&1.148505906423&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.032100891654&{}:{}&0.404105755843&{}:{}&0.627995135811&. \end{alignedat} \]
5c (312)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5c}}&{}\approx{}&-0.109217382054&{}:{}&-0.160117916057&{}:{}&1.269335298110&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-1.980750764737&{}:{}&-35.769009176847&{}:{}&38.749759941585&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.305524282344&{}:{}&0.511161148250&{}:{}&0.794363134093&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.061771066627&{}:{}&-0.148250559905&{}:{}&1.210021626533&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.102765041016&{}:{}&-0.037351893006&{}:{}&1.140116934021&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.069691765945&{}:{}&-0.295277294348&{}:{}&1.364969060293&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.339176265378&{}:{}&-0.217635637804&{}:{}&1.556811903183&, \end{alignedat} \]
5c (312)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5c}}&{}\approx{}&-0.705254708953&{}:{}&0.309427494852&{}:{}&1.395827214101&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.181455294172&{}:{}&0.818544705828&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.021260915682&{}:{}&0.000000000000&{}:{}&2.021260915682&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.781723650696&{}:{}&-0.781723650696&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5c}}&{}\approx{}&-0.204286721104&{}:{}&-0.253511717770&{}:{}&1.457798438875&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.102765041016&{}:{}&-0.037351893006&{}:{}&1.140116934021&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.069691765945&{}:{}&-0.295277294348&{}:{}&1.364969060293&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.339176265378&{}:{}&-0.217635637804&{}:{}&1.556811903183&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.181455294172&{}:{}&0.818544705828&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.021260915682&{}:{}&0.000000000000&{}:{}&2.021260915682&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.781723650696&{}:{}&-0.781723650696&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)