Derousseau's Generalization of the Malfatti circles

Test Case in ETC

The test case used in “Encyclopedis of Triangle Centers” to search for triangle centers.

\(a:b:c=6:9:13\).

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5b (303)

Malfatti circles

5b (303)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&, \\ P_{\mathbf{5b}}&{}\approx{}&0.014088764253&{}:{}&-0.003506974885&{}:{}&0.989418210631&, \\ P^-_{\mathbf{5b}}&{}\approx{}&-0.006737440621&{}:{}&0.028358854392&{}:{}&0.978378586229&, \\ P^+_{\mathbf{5b}}&{}\approx{}&0.033532700373&{}:{}&-0.033257817757&{}:{}&0.999725117384&, \\ Q_{\mathbf{5b}}&{}\approx{}&0.244129673288&{}:{}&0.032552656001&{}:{}&0.723317670710&, \\ I^\prime_{\mathbf{5b}}&{}\approx{}&0.078590112147&{}:{}&-0.048022350675&{}:{}&0.969432238529&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{5b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{5b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{5b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{5b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{5b}}\) Radical center of the Malfatti circles
5b (303)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5b}}&{}\approx{}&-0.900660823394&{}:{}&-4.276486852636&{}:{}&6.177147676029&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.222400965174&{}:{}&0.295730276948&{}:{}&0.481868757878&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.022038500742&{}:{}&-0.033057751114&{}:{}&1.011019250371&, \\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.040103048153&{}:{}&-0.003414439738&{}:{}&0.963311391585&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.014960638134&{}:{}&-0.065608341954&{}:{}&1.050647703820&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.141132448352&{}:{}&-0.035130685905&{}:{}&0.893998237553&, \\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.020744798798&{}:{}&0.027584705349&{}:{}&0.951670495854&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.007170154719&{}:{}&-0.034045082888&{}:{}&1.041215237607&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.099328488719&{}:{}&0.418087862595&{}:{}&0.681240626124&, \\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.043066455781&{}:{}&0.956933544219&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.252344145449&{}:{}&0.000000000000&{}:{}&0.747655854551&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.882346458175&{}:{}&0.117653541825&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5b}}}{B^\prime_{\mathbf{5b}}}{C^\prime_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^*_{\mathbf{5b}}}{B^*_{\mathbf{5b}}}{C^*_{\mathbf{5b}}}\)
5b (303)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5b}}}}&{}\approx{}&4.751652058484&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5b}}}}&{}\approx{}&0.370668275290&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5b}}}}&{}\approx{}&0.036730834571&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&-0.900660823394&{}:{}&-4.276486852636&{}:{}&6.177147676029&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.222400965174&{}:{}&0.295730276948&{}:{}&0.481868757878&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.022038500742&{}:{}&-0.033057751114&{}:{}&1.011019250371&. \end{alignedat} \]
5b (303)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5b}}&{}\approx{}&0.014088764253&{}:{}&-0.003506974885&{}:{}&0.989418210631&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.040103048153&{}:{}&-0.003414439738&{}:{}&0.963311391585&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.014960638134&{}:{}&-0.065608341954&{}:{}&1.050647703820&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.141132448352&{}:{}&-0.035130685905&{}:{}&0.893998237553&. \end{alignedat} \]
5b (303)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5b}}&{}\approx{}&-0.006737440621&{}:{}&0.028358854392&{}:{}&0.978378586229&,\\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.020744798798&{}:{}&0.027584705349&{}:{}&0.951670495854&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.007170154719&{}:{}&-0.034045082888&{}:{}&1.041215237607&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.099328488719&{}:{}&0.418087862595&{}:{}&0.681240626124&. \end{alignedat} \]
5b (303)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5b}}&{}\approx{}&0.033532700373&{}:{}&-0.033257817757&{}:{}&0.999725117384&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&-0.900660823394&{}:{}&-4.276486852636&{}:{}&6.177147676029&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.222400965174&{}:{}&0.295730276948&{}:{}&0.481868757878&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.022038500742&{}:{}&-0.033057751114&{}:{}&1.011019250371&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.040103048153&{}:{}&-0.003414439738&{}:{}&0.963311391585&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.014960638134&{}:{}&-0.065608341954&{}:{}&1.050647703820&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.141132448352&{}:{}&-0.035130685905&{}:{}&0.893998237553&, \end{alignedat} \]
5b (303)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5b}}&{}\approx{}&0.244129673288&{}:{}&0.032552656001&{}:{}&0.723317670710&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.043066455781&{}:{}&0.956933544219&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.252344145449&{}:{}&0.000000000000&{}:{}&0.747655854551&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.882346458175&{}:{}&0.117653541825&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5b}}&{}\approx{}&0.078590112147&{}:{}&-0.048022350675&{}:{}&0.969432238529&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.040103048153&{}:{}&-0.003414439738&{}:{}&0.963311391585&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.014960638134&{}:{}&-0.065608341954&{}:{}&1.050647703820&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.141132448352&{}:{}&-0.035130685905&{}:{}&0.893998237553&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.043066455781&{}:{}&0.956933544219&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.252344145449&{}:{}&0.000000000000&{}:{}&0.747655854551&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.882346458175&{}:{}&0.117653541825&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)