Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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0c(110)

Malfatti circles

0c (110)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&, \\ P_{\mathbf{c}}&{}\approx{}&0.874017676303&{}:{}&1.241608990642&{}:{}&-1.115626666945&, \\ P^-_{\mathbf{c}}&{}\approx{}&0.803698348760&{}:{}&0.912740143513&{}:{}&-0.716438492273&, \\ P^+_{\mathbf{c}}&{}\approx{}&0.925194119490&{}:{}&1.480950555180&{}:{}&-1.406144674671&, \\ Q_{\mathbf{c}}&{}\approx{}&0.871414193625&{}:{}&0.568875949233&{}:{}&-0.440290142859&, \\ I^\prime_{\mathbf{c}}&{}\approx{}&0.975882762576&{}:{}&1.801920325713&{}:{}&-1.777803088289&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{c}}\)
\(P^-_{\mathbf{c}}\)
\(P^+_{\mathbf{c}}\)
\(Q_{\mathbf{c}}\)
\(I^\prime_{\mathbf{c}}\)
0c (110)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{c}}&{}\approx{}&1.061771066627&{}:{}&0.741252799527&{}:{}&-0.803023866155&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.433993829509&{}:{}&1.694390127214&{}:{}&-1.128383956722&,\\C^\prime_{\mathbf{c}}&{}\approx{}&2.980750764737&{}:{}&7.153801835369&{}:{}&-9.134552600107&, \\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.757671564297&{}:{}&2.388249046583&{}:{}&-2.145920610881&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.241940696762&{}:{}&1.343315754724&{}:{}&-1.585256451486&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.800759333081&{}:{}&1.137539908231&{}:{}&-0.938299241311&, \\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.543576880205&{}:{}&2.122222106807&{}:{}&-1.665798987012&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.239615826008&{}:{}&0.865411320997&{}:{}&-1.105027147005&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.724969175496&{}:{}&0.823329387582&{}:{}&-0.548298563079&, \\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&4.424095981292&{}:{}&-3.424095981292&,\\B^*_{\mathbf{c}}&{}\approx{}&2.021260915682&{}:{}&0.000000000000&{}:{}&-1.021260915682&,\\C^*_{\mathbf{c}}&{}\approx{}&0.605026839867&{}:{}&0.394973160133&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{c}}}{B^\prime_{\mathbf{c}}}{C^\prime_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{c}}}{B^{\prime\prime}_{\mathbf{c}}}{C^{\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{c}}}{B^{\prime\prime\prime}_{\mathbf{c}}}{C^{\prime\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^*_{\mathbf{c}}}{B^*_{\mathbf{c}}}{C^*_{\mathbf{c}}}\)
0c (110)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{c}}}}&{}\approx{}&0.247084266509&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{c}}}}&{}\approx{}&0.347195063607&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{c}}}}&{}\approx{}&2.384600611790&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.061771066627&{}:{}&0.741252799527&{}:{}&-0.803023866155&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.433993829509&{}:{}&1.694390127214&{}:{}&-1.128383956722&,\\C^\prime_{\mathbf{c}}&{}\approx{}&2.980750764737&{}:{}&7.153801835369&{}:{}&-9.134552600107&. \end{alignedat} \]
0c (110)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{c}}&{}\approx{}&0.874017676303&{}:{}&1.241608990642&{}:{}&-1.115626666945&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.757671564297&{}:{}&2.388249046583&{}:{}&-2.145920610881&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.241940696762&{}:{}&1.343315754724&{}:{}&-1.585256451486&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.800759333081&{}:{}&1.137539908231&{}:{}&-0.938299241311&. \end{alignedat} \]
0c (110)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{c}}&{}\approx{}&0.803698348760&{}:{}&0.912740143513&{}:{}&-0.716438492273&,\\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.543576880205&{}:{}&2.122222106807&{}:{}&-1.665798987012&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.239615826008&{}:{}&0.865411320997&{}:{}&-1.105027147005&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.724969175496&{}:{}&0.823329387582&{}:{}&-0.548298563079&. \end{alignedat} \]
0c (110)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{c}}&{}\approx{}&0.925194119490&{}:{}&1.480950555180&{}:{}&-1.406144674671&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.061771066627&{}:{}&0.741252799527&{}:{}&-0.803023866155&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.433993829509&{}:{}&1.694390127214&{}:{}&-1.128383956722&,\\C^\prime_{\mathbf{c}}&{}\approx{}&2.980750764737&{}:{}&7.153801835369&{}:{}&-9.134552600107&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.757671564297&{}:{}&2.388249046583&{}:{}&-2.145920610881&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.241940696762&{}:{}&1.343315754724&{}:{}&-1.585256451486&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.800759333081&{}:{}&1.137539908231&{}:{}&-0.938299241311&, \end{alignedat} \]
0c (110)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{c}}&{}\approx{}&0.871414193625&{}:{}&0.568875949233&{}:{}&-0.440290142859&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&4.424095981292&{}:{}&-3.424095981292&,\\B^*_{\mathbf{c}}&{}\approx{}&2.021260915682&{}:{}&0.000000000000&{}:{}&-1.021260915682&,\\C^*_{\mathbf{c}}&{}\approx{}&0.605026839867&{}:{}&0.394973160133&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{c}}&{}\approx{}&0.975882762576&{}:{}&1.801920325713&{}:{}&-1.777803088289&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.757671564297&{}:{}&2.388249046583&{}:{}&-2.145920610881&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.241940696762&{}:{}&1.343315754724&{}:{}&-1.585256451486&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.800759333081&{}:{}&1.137539908231&{}:{}&-0.938299241311&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&4.424095981292&{}:{}&-3.424095981292&,\\B^*_{\mathbf{c}}&{}\approx{}&2.021260915682&{}:{}&0.000000000000&{}:{}&-1.021260915682&,\\C^*_{\mathbf{c}}&{}\approx{}&0.605026839867&{}:{}&0.394973160133&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)