Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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1b(103)

Malfatti circles

1b (103)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&, \\ P_{\mathbf{1b}}&{}\approx{}&0.914335235943&{}:{}&-0.158822226047&{}:{}&0.244486990104&, \\ P^-_{\mathbf{1b}}&{}\approx{}&0.938648316351&{}:{}&-0.034371600845&{}:{}&0.095723284494&, \\ P^+_{\mathbf{1b}}&{}\approx{}&0.895573639399&{}:{}&-0.254856639641&{}:{}&0.359283000241&, \\ Q_{\mathbf{1b}}&{}\approx{}&1.147315378081&{}:{}&0.195213613534&{}:{}&-0.342528991616&, \\ I^\prime_{\mathbf{1b}}&{}\approx{}&0.842941130372&{}:{}&-0.405666854564&{}:{}&0.562725724193&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{1b}}\)
\(P^-_{\mathbf{1b}}\)
\(P^+_{\mathbf{1b}}\)
\(Q_{\mathbf{1b}}\)
\(I^\prime_{\mathbf{1b}}\)
1b (103)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1b}}&{}\approx{}&0.962089301849&{}:{}&-0.151642792606&{}:{}&0.189553490757&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.424285434966&{}:{}&-0.131427826575&{}:{}&0.707142391609&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.889140284973&{}:{}&-2.518853713298&{}:{}&1.629713428324&, \\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.692941814049&{}:{}&-0.569284993142&{}:{}&0.876343179093&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.014731422764&{}:{}&-0.286063666742&{}:{}&0.271332243978&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.848402373744&{}:{}&-0.147369529561&{}:{}&0.298967155817&, \\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.643951538543&{}:{}&-0.199472204690&{}:{}&0.555520666147&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.058916285427&{}:{}&-0.166904488335&{}:{}&0.107988202908&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.861768406823&{}:{}&-0.031556397838&{}:{}&0.169787991015&, \\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.325140769936&{}:{}&2.325140769936&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.425614793411&{}:{}&0.000000000000&{}:{}&-0.425614793411&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.854592627233&{}:{}&0.145407372767&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1b}}}{B^\prime_{\mathbf{1b}}}{C^\prime_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^*_{\mathbf{1b}}}{B^*_{\mathbf{1b}}}{C^*_{\mathbf{1b}}}\)
1b (103)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1b}}}}&{}\approx{}&0.151642792606&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1b}}}}&{}\approx{}&0.565713913288&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1b}}}}&{}\approx{}&2.518853713298&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.962089301849&{}:{}&-0.151642792606&{}:{}&0.189553490757&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.424285434966&{}:{}&-0.131427826575&{}:{}&0.707142391609&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.889140284973&{}:{}&-2.518853713298&{}:{}&1.629713428324&. \end{alignedat} \]
1b (103)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1b}}&{}\approx{}&0.914335235943&{}:{}&-0.158822226047&{}:{}&0.244486990104&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.692941814049&{}:{}&-0.569284993142&{}:{}&0.876343179093&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.014731422764&{}:{}&-0.286063666742&{}:{}&0.271332243978&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.848402373744&{}:{}&-0.147369529561&{}:{}&0.298967155817&. \end{alignedat} \]
1b (103)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1b}}&{}\approx{}&0.938648316351&{}:{}&-0.034371600845&{}:{}&0.095723284494&,\\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.643951538543&{}:{}&-0.199472204690&{}:{}&0.555520666147&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.058916285427&{}:{}&-0.166904488335&{}:{}&0.107988202908&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.861768406823&{}:{}&-0.031556397838&{}:{}&0.169787991015&. \end{alignedat} \]
1b (103)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1b}}&{}\approx{}&0.895573639399&{}:{}&-0.254856639641&{}:{}&0.359283000241&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.962089301849&{}:{}&-0.151642792606&{}:{}&0.189553490757&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.424285434966&{}:{}&-0.131427826575&{}:{}&0.707142391609&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.889140284973&{}:{}&-2.518853713298&{}:{}&1.629713428324&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.692941814049&{}:{}&-0.569284993142&{}:{}&0.876343179093&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.014731422764&{}:{}&-0.286063666742&{}:{}&0.271332243978&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.848402373744&{}:{}&-0.147369529561&{}:{}&0.298967155817&, \end{alignedat} \]
1b (103)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1b}}&{}\approx{}&1.147315378081&{}:{}&0.195213613534&{}:{}&-0.342528991616&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.325140769936&{}:{}&2.325140769936&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.425614793411&{}:{}&0.000000000000&{}:{}&-0.425614793411&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.854592627233&{}:{}&0.145407372767&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1b}}&{}\approx{}&0.842941130372&{}:{}&-0.405666854564&{}:{}&0.562725724193&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.692941814049&{}:{}&-0.569284993142&{}:{}&0.876343179093&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.014731422764&{}:{}&-0.286063666742&{}:{}&0.271332243978&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.848402373744&{}:{}&-0.147369529561&{}:{}&0.298967155817&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.325140769936&{}:{}&2.325140769936&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.425614793411&{}:{}&0.000000000000&{}:{}&-0.425614793411&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.854592627233&{}:{}&0.145407372767&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)