Derousseau's Generalization of the Malfatti circles

Test Case in ETC

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

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


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\(\mathbf{7c}\) \((332)\)

Malfatti circles

7c (332)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&, \\ P_{\mathbf{7c}}&{}\approx{}&-0.092222142236&{}:{}&-0.115564984468&{}:{}&1.207787126704&, \\ P^-_{\mathbf{7c}}&{}\approx{}&-0.006177878347&{}:{}&0.012867861963&{}:{}&0.993310016384&, \\ P^+_{\mathbf{7c}}&{}\approx{}&-0.183337143514&{}:{}&-0.251566600817&{}:{}&1.434903744331&, \\ Q_{\mathbf{7c}}&{}\approx{}&0.247054145389&{}:{}&0.218639025809&{}:{}&0.534306828802&, \\ I^\prime_{\mathbf{7c}}&{}\approx{}&-0.338309036996&{}:{}&-0.463825645820&{}:{}&1.802134682816&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{7c}}\)
\(P^-_{\mathbf{7c}}\)
\(P^+_{\mathbf{7c}}\)
\(Q_{\mathbf{7c}}\)
\(I^\prime_{\mathbf{7c}}\)
7c (332)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7c}}&{}\approx{}&-0.124870757207&{}:{}&-2.530959203716&{}:{}&3.655829960922&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.757109304120&{}:{}&0.116705811860&{}:{}&1.640403492259&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.106393721105&{}:{}&-0.159590581657&{}:{}&1.265984302762&, \\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.186569528526&{}:{}&-0.125547618778&{}:{}&1.312117147303&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.107489690767&{}:{}&-0.300248858926&{}:{}&1.407738549693&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.561285927098&{}:{}&-0.703356025722&{}:{}&2.264641952821&, \\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.090471558268&{}:{}&0.013945881527&{}:{}&1.076525676741&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.007167618967&{}:{}&-0.145277818439&{}:{}&1.152445437407&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.032938289373&{}:{}&0.068606945157&{}:{}&0.964331344216&, \\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.290378151988&{}:{}&0.709621848012&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.316184393064&{}:{}&0.000000000000&{}:{}&0.683815606936&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.530508413411&{}:{}&0.469491586589&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7c}}}{B^\prime_{\mathbf{7c}}}{C^\prime_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^*_{\mathbf{7c}}}{B^*_{\mathbf{7c}}}{C^*_{\mathbf{7c}}}\)
7c (332)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.562435378603&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.252369768040&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.035464573702&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&-0.124870757207&{}:{}&-2.530959203716&{}:{}&3.655829960922&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.757109304120&{}:{}&0.116705811860&{}:{}&1.640403492259&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.106393721105&{}:{}&-0.159590581657&{}:{}&1.265984302762&. \end{alignedat} \]
7c (332)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7c}}&{}\approx{}&-0.092222142236&{}:{}&-0.115564984468&{}:{}&1.207787126704&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.186569528526&{}:{}&-0.125547618778&{}:{}&1.312117147303&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.107489690767&{}:{}&-0.300248858926&{}:{}&1.407738549693&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.561285927098&{}:{}&-0.703356025722&{}:{}&2.264641952821&. \end{alignedat} \]
7c (332)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7c}}&{}\approx{}&-0.006177878347&{}:{}&0.012867861963&{}:{}&0.993310016384&,\\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.090471558268&{}:{}&0.013945881527&{}:{}&1.076525676741&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.007167618967&{}:{}&-0.145277818439&{}:{}&1.152445437407&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.032938289373&{}:{}&0.068606945157&{}:{}&0.964331344216&. \end{alignedat} \]
7c (332)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7c}}&{}\approx{}&-0.183337143514&{}:{}&-0.251566600817&{}:{}&1.434903744331&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&-0.124870757207&{}:{}&-2.530959203716&{}:{}&3.655829960922&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.757109304120&{}:{}&0.116705811860&{}:{}&1.640403492259&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.106393721105&{}:{}&-0.159590581657&{}:{}&1.265984302762&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.186569528526&{}:{}&-0.125547618778&{}:{}&1.312117147303&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.107489690767&{}:{}&-0.300248858926&{}:{}&1.407738549693&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.561285927098&{}:{}&-0.703356025722&{}:{}&2.264641952821&, \end{alignedat} \]
7c (332)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7c}}&{}\approx{}&0.247054145389&{}:{}&0.218639025809&{}:{}&0.534306828802&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.290378151988&{}:{}&0.709621848012&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.316184393064&{}:{}&0.000000000000&{}:{}&0.683815606936&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.530508413411&{}:{}&0.469491586589&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7c}}&{}\approx{}&-0.338309036996&{}:{}&-0.463825645820&{}:{}&1.802134682816&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.186569528526&{}:{}&-0.125547618778&{}:{}&1.312117147303&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.107489690767&{}:{}&-0.300248858926&{}:{}&1.407738549693&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.561285927098&{}:{}&-0.703356025722&{}:{}&2.264641952821&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.290378151988&{}:{}&0.709621848012&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.316184393064&{}:{}&0.000000000000&{}:{}&0.683815606936&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.530508413411&{}:{}&0.469491586589&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Hiroyasu Kamo