Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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1c(112)

Malfatti circles

1c (112)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&, \\ P_{\mathbf{1c}}&{}\approx{}&-0.092900105345&{}:{}&-0.726622872365&{}:{}&1.819522977710&, \\ P^-_{\mathbf{1c}}&{}\approx{}&0.157356793669&{}:{}&-0.032145986020&{}:{}&0.874789192351&, \\ P^+_{\mathbf{1c}}&{}\approx{}&-0.491850103855&{}:{}&-1.833731423075&{}:{}&3.325581526930&, \\ Q_{\mathbf{1c}}&{}\approx{}&1.809509648629&{}:{}&2.771838734689&{}:{}&-3.581348383318&, \\ I^\prime_{\mathbf{1c}}&{}\approx{}&-0.554506108829&{}:{}&-2.402472276392&{}:{}&3.956978385220&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{1c}}\)
\(P^-_{\mathbf{1c}}\)
\(P^+_{\mathbf{1c}}\)
\(Q_{\mathbf{1c}}\)
\(I^\prime_{\mathbf{1c}}\)
1c (112)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1c}}&{}\approx{}&0.771562266706&{}:{}&-2.741252799527&{}:{}&2.969690532821&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-0.683993829509&{}:{}&-0.094390127214&{}:{}&1.778383956722&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-2.814084098071&{}:{}&-6.753801835369&{}:{}&10.567885933440&, \\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.100499301987&{}:{}&-1.396532792654&{}:{}&3.497032094641&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.263625366595&{}:{}&-3.899688606449&{}:{}&5.163313973044&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.138817544824&{}:{}&-1.085768447511&{}:{}&2.224585992335&, \\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.382064822711&{}:{}&-0.052724375080&{}:{}&1.434789197791&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.332622437876&{}:{}&-1.181760991121&{}:{}&1.849138553245&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.215301729012&{}:{}&-0.043983397282&{}:{}&0.828681668269&, \\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-3.424095981292&{}:{}&4.424095981292&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.021260915682&{}:{}&0.000000000000&{}:{}&2.021260915682&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.394973160133&{}:{}&0.605026839867&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1c}}}{B^\prime_{\mathbf{1c}}}{C^\prime_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^*_{\mathbf{1c}}}{B^*_{\mathbf{1c}}}{C^*_{\mathbf{1c}}}\)
1c (112)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.913750933176&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.547195063607&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1c}}}}&{}\approx{}&-2.251267278456&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.771562266706&{}:{}&-2.741252799527&{}:{}&2.969690532821&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-0.683993829509&{}:{}&-0.094390127214&{}:{}&1.778383956722&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-2.814084098071&{}:{}&-6.753801835369&{}:{}&10.567885933440&. \end{alignedat} \]
1c (112)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1c}}&{}\approx{}&-0.092900105345&{}:{}&-0.726622872365&{}:{}&1.819522977710&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.100499301987&{}:{}&-1.396532792654&{}:{}&3.497032094641&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.263625366595&{}:{}&-3.899688606449&{}:{}&5.163313973044&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.138817544824&{}:{}&-1.085768447511&{}:{}&2.224585992335&. \end{alignedat} \]
1c (112)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1c}}&{}\approx{}&0.157356793669&{}:{}&-0.032145986020&{}:{}&0.874789192351&,\\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.382064822711&{}:{}&-0.052724375080&{}:{}&1.434789197791&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.332622437876&{}:{}&-1.181760991121&{}:{}&1.849138553245&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.215301729012&{}:{}&-0.043983397282&{}:{}&0.828681668269&. \end{alignedat} \]
1c (112)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1c}}&{}\approx{}&-0.491850103855&{}:{}&-1.833731423075&{}:{}&3.325581526930&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.771562266706&{}:{}&-2.741252799527&{}:{}&2.969690532821&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-0.683993829509&{}:{}&-0.094390127214&{}:{}&1.778383956722&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-2.814084098071&{}:{}&-6.753801835369&{}:{}&10.567885933440&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.100499301987&{}:{}&-1.396532792654&{}:{}&3.497032094641&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.263625366595&{}:{}&-3.899688606449&{}:{}&5.163313973044&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.138817544824&{}:{}&-1.085768447511&{}:{}&2.224585992335&, \end{alignedat} \]
1c (112)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1c}}&{}\approx{}&1.809509648629&{}:{}&2.771838734689&{}:{}&-3.581348383318&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-3.424095981292&{}:{}&4.424095981292&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.021260915682&{}:{}&0.000000000000&{}:{}&2.021260915682&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.394973160133&{}:{}&0.605026839867&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1c}}&{}\approx{}&-0.554506108829&{}:{}&-2.402472276392&{}:{}&3.956978385220&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.100499301987&{}:{}&-1.396532792654&{}:{}&3.497032094641&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.263625366595&{}:{}&-3.899688606449&{}:{}&5.163313973044&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.138817544824&{}:{}&-1.085768447511&{}:{}&2.224585992335&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-3.424095981292&{}:{}&4.424095981292&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.021260915682&{}:{}&0.000000000000&{}:{}&2.021260915682&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.394973160133&{}:{}&0.605026839867&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)