Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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4a(211)

Malfatti circles

4a (211)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&, \\ P_{\mathbf{4a}}&{}\approx{}&1.016253645879&{}:{}&-0.004638489525&{}:{}&-0.011615156354&, \\ P^-_{\mathbf{4a}}&{}\approx{}&0.722078621358&{}:{}&0.135830636120&{}:{}&0.142090742521&, \\ P^+_{\mathbf{4a}}&{}\approx{}&1.565742096837&{}:{}&-0.267020254885&{}:{}&-0.298721841952&, \\ Q_{\mathbf{4a}}&{}\approx{}&1.736333899111&{}:{}&-0.321255896499&{}:{}&-0.415078002612&, \\ I^\prime_{\mathbf{4a}}&{}\approx{}&1.383216537776&{}:{}&-0.144771483710&{}:{}&-0.238445054066&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{4a}}\)
\(P^-_{\mathbf{4a}}\)
\(P^+_{\mathbf{4a}}\)
\(Q_{\mathbf{4a}}\)
\(I^\prime_{\mathbf{4a}}\)
4a (211)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4a}}&{}\approx{}&1.672718770973&{}:{}&-0.322905010067&{}:{}&-0.349813760906&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.353748070570&{}:{}&3.165996912912&{}:{}&-3.519744983482&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.955589498018&{}:{}&-2.293414795243&{}:{}&2.337825297225&, \\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.120345034693&{}:{}&-0.034344244176&{}:{}&-0.086000790517&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.584210999625&{}:{}&-0.566104438795&{}:{}&-0.018106560830&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.643883353889&{}:{}&-0.007503181660&{}:{}&-0.636380172229&, \\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.172855936653&{}:{}&0.404256429774&{}:{}&0.422887633572&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.996275787094&{}:{}&-0.192323090195&{}:{}&0.196047303101&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.021469515834&{}:{}&0.192149234182&{}:{}&-0.213618750017&, \\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.436291058834&{}:{}&0.563708941166&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.314154134496&{}:{}&0.000000000000&{}:{}&-0.314154134496&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.227023454471&{}:{}&-0.227023454471&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4a}}}{B^\prime_{\mathbf{4a}}}{C^\prime_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^*_{\mathbf{4a}}}{B^*_{\mathbf{4a}}}{C^*_{\mathbf{4a}}}\)
4a (211)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4a}}}}&{}\approx{}&-0.538175016779&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4a}}}}&{}\approx{}&-5.414992282281&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4a}}}}&{}\approx{}&-3.822357992072&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&1.672718770973&{}:{}&-0.322905010067&{}:{}&-0.349813760906&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.353748070570&{}:{}&3.165996912912&{}:{}&-3.519744983482&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.955589498018&{}:{}&-2.293414795243&{}:{}&2.337825297225&. \end{alignedat} \]
4a (211)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4a}}&{}\approx{}&1.016253645879&{}:{}&-0.004638489525&{}:{}&-0.011615156354&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.120345034693&{}:{}&-0.034344244176&{}:{}&-0.086000790517&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.584210999625&{}:{}&-0.566104438795&{}:{}&-0.018106560830&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.643883353889&{}:{}&-0.007503181660&{}:{}&-0.636380172229&. \end{alignedat} \]
4a (211)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4a}}&{}\approx{}&0.722078621358&{}:{}&0.135830636120&{}:{}&0.142090742521&,\\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.172855936653&{}:{}&0.404256429774&{}:{}&0.422887633572&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.996275787094&{}:{}&-0.192323090195&{}:{}&0.196047303101&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.021469515834&{}:{}&0.192149234182&{}:{}&-0.213618750017&. \end{alignedat} \]
4a (211)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4a}}&{}\approx{}&1.565742096837&{}:{}&-0.267020254885&{}:{}&-0.298721841952&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&1.672718770973&{}:{}&-0.322905010067&{}:{}&-0.349813760906&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.353748070570&{}:{}&3.165996912912&{}:{}&-3.519744983482&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.955589498018&{}:{}&-2.293414795243&{}:{}&2.337825297225&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.120345034693&{}:{}&-0.034344244176&{}:{}&-0.086000790517&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.584210999625&{}:{}&-0.566104438795&{}:{}&-0.018106560830&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.643883353889&{}:{}&-0.007503181660&{}:{}&-0.636380172229&, \end{alignedat} \]
4a (211)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4a}}&{}\approx{}&1.736333899111&{}:{}&-0.321255896499&{}:{}&-0.415078002612&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.436291058834&{}:{}&0.563708941166&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.314154134496&{}:{}&0.000000000000&{}:{}&-0.314154134496&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.227023454471&{}:{}&-0.227023454471&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4a}}&{}\approx{}&1.383216537776&{}:{}&-0.144771483710&{}:{}&-0.238445054066&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.120345034693&{}:{}&-0.034344244176&{}:{}&-0.086000790517&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.584210999625&{}:{}&-0.566104438795&{}:{}&-0.018106560830&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.643883353889&{}:{}&-0.007503181660&{}:{}&-0.636380172229&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.436291058834&{}:{}&0.563708941166&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.314154134496&{}:{}&0.000000000000&{}:{}&-0.314154134496&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.227023454471&{}:{}&-0.227023454471&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)