Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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6a(231)

Malfatti circles

6a (231)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&, \\ P_{\mathbf{6a}}&{}\approx{}&1.004924888365&{}:{}&-0.004586781644&{}:{}&-0.000338106721&, \\ P^-_{\mathbf{6a}}&{}\approx{}&0.943639397878&{}:{}&0.024938808133&{}:{}&0.031421793988&, \\ P^+_{\mathbf{6a}}&{}\approx{}&1.072844190630&{}:{}&-0.037308351475&{}:{}&-0.035535839155&, \\ Q_{\mathbf{6a}}&{}\approx{}&1.168230342340&{}:{}&-0.216145573232&{}:{}&0.047915230892&, \\ I^\prime_{\mathbf{6a}}&{}\approx{}&1.155053746175&{}:{}&-0.120891299397&{}:{}&-0.034162446779&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{6a}}\)
\(P^-_{\mathbf{6a}}\)
\(P^+_{\mathbf{6a}}\)
\(Q_{\mathbf{6a}}\)
\(I^\prime_{\mathbf{6a}}\)
6a (231)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6a}}&{}\approx{}&1.069165380380&{}:{}&-0.033199382582&{}:{}&-0.035965997798&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&2.259078156460&{}:{}&4.614525050336&{}:{}&-5.873603206796&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.852036107425&{}:{}&-2.044886657820&{}:{}&2.192850550395&, \\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.237380562420&{}:{}&-0.221083753689&{}:{}&-0.016296808731&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.065696535654&{}:{}&-0.065337982327&{}:{}&-0.000358553327&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.076407261946&{}:{}&-0.004913048853&{}:{}&-0.071494213093&, \\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.178052608217&{}:{}&0.363700661948&{}:{}&0.458246729835&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.997758199656&{}:{}&-0.030982069568&{}:{}&0.033223869913&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.007263373092&{}:{}&0.026620283191&{}:{}&-0.033883656283&, \\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.284819196258&{}:{}&-0.284819196258&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.960600743902&{}:{}&0.000000000000&{}:{}&0.039399256098&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.227023454471&{}:{}&-0.227023454471&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6a}}}{B^\prime_{\mathbf{6a}}}{C^\prime_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^*_{\mathbf{6a}}}{B^*_{\mathbf{6a}}}{C^*_{\mathbf{6a}}}\)
6a (231)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6a}}}}&{}\approx{}&-0.055332304304&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6a}}}}&{}\approx{}&-9.036312625840&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6a}}}}&{}\approx{}&-3.408144429699&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.250000000000&{}:{}&0.600000000000&{}:{}&0.650000000000&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.069165380380&{}:{}&-0.033199382582&{}:{}&-0.035965997798&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&2.259078156460&{}:{}&4.614525050336&{}:{}&-5.873603206796&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.852036107425&{}:{}&-2.044886657820&{}:{}&2.192850550395&. \end{alignedat} \]
6a (231)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6a}}&{}\approx{}&1.004924888365&{}:{}&-0.004586781644&{}:{}&-0.000338106721&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.237380562420&{}:{}&-0.221083753689&{}:{}&-0.016296808731&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.065696535654&{}:{}&-0.065337982327&{}:{}&-0.000358553327&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.076407261946&{}:{}&-0.004913048853&{}:{}&-0.071494213093&. \end{alignedat} \]
6a (231)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6a}}&{}\approx{}&0.943639397878&{}:{}&0.024938808133&{}:{}&0.031421793988&,\\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.178052608217&{}:{}&0.363700661948&{}:{}&0.458246729835&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.997758199656&{}:{}&-0.030982069568&{}:{}&0.033223869913&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.007263373092&{}:{}&0.026620283191&{}:{}&-0.033883656283&. \end{alignedat} \]
6a (231)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6a}}&{}\approx{}&1.072844190630&{}:{}&-0.037308351475&{}:{}&-0.035535839155&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.069165380380&{}:{}&-0.033199382582&{}:{}&-0.035965997798&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&2.259078156460&{}:{}&4.614525050336&{}:{}&-5.873603206796&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.852036107425&{}:{}&-2.044886657820&{}:{}&2.192850550395&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.237380562420&{}:{}&-0.221083753689&{}:{}&-0.016296808731&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.065696535654&{}:{}&-0.065337982327&{}:{}&-0.000358553327&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.076407261946&{}:{}&-0.004913048853&{}:{}&-0.071494213093&, \end{alignedat} \]
6a (231)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6a}}&{}\approx{}&1.168230342340&{}:{}&-0.216145573232&{}:{}&0.047915230892&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.284819196258&{}:{}&-0.284819196258&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.960600743902&{}:{}&0.000000000000&{}:{}&0.039399256098&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.227023454471&{}:{}&-0.227023454471&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6a}}&{}\approx{}&1.155053746175&{}:{}&-0.120891299397&{}:{}&-0.034162446779&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.237380562420&{}:{}&-0.221083753689&{}:{}&-0.016296808731&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.065696535654&{}:{}&-0.065337982327&{}:{}&-0.000358553327&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.076407261946&{}:{}&-0.004913048853&{}:{}&-0.071494213093&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.284819196258&{}:{}&-0.284819196258&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.960600743902&{}:{}&0.000000000000&{}:{}&0.039399256098&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.227023454471&{}:{}&-0.227023454471&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)