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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\(\mathbf{1a}\) \((013)\)

Malfatti circles

1a (013)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&, \\ P_{\mathbf{1a}}&{}\approx{}&-0.070640817164&{}:{}&0.897603315436&{}:{}&0.173037501728&, \\ P^-_{\mathbf{1a}}&{}\approx{}&-0.018051002358&{}:{}&0.955505367660&{}:{}&0.062545634699&, \\ P^+_{\mathbf{1a}}&{}\approx{}&-0.109724275369&{}:{}&0.854571932906&{}:{}&0.255152342463&, \\ Q_{\mathbf{1a}}&{}\approx{}&0.085333645441&{}:{}&1.138147853490&{}:{}&-0.223481498931&, \\ I^\prime_{\mathbf{1a}}&{}\approx{}&-0.176381553416&{}:{}&0.770039776868&{}:{}&0.406341776547&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{1a}}\)
\(P^-_{\mathbf{1a}}\)
\(P^+_{\mathbf{1a}}\)
\(Q_{\mathbf{1a}}\)
\(I^\prime_{\mathbf{1a}}\)
1a (013)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1a}}&{}\approx{}&-0.082224400214&{}:{}&0.442728163724&{}:{}&0.639496236490&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.062841355180&{}:{}&0.926685085623&{}:{}&0.136156269557&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.682466322424&{}:{}&1.023699483636&{}:{}&0.658766838788&, \\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.115085648128&{}:{}&0.934864950700&{}:{}&0.180220697428&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.263443026642&{}:{}&0.618128815849&{}:{}&0.645314210793&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.066243810640&{}:{}&0.841732392764&{}:{}&0.224511417877&, \\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.067973133574&{}:{}&1.002360450786&{}:{}&0.065612682789&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.127399428194&{}:{}&0.685968091671&{}:{}&0.441431336523&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.016747305823&{}:{}&0.886495956863&{}:{}&0.130251348960&, \\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.244331168209&{}:{}&-0.244331168209&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.617697946698&{}:{}&0.000000000000&{}:{}&1.617697946698&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.069746576075&{}:{}&0.930253423925&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1a}}}{B^\prime_{\mathbf{1a}}}{C^\prime_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^*_{\mathbf{1a}}}{B^*_{\mathbf{1a}}}{C^*_{\mathbf{1a}}}\)
1a (013)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1a}}}}&{}\approx{}&0.787072291065&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1a}}}}&{}\approx{}&0.167576947147&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1a}}}}&{}\approx{}&1.819910193131&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.082224400214&{}:{}&0.442728163724&{}:{}&0.639496236490&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.062841355180&{}:{}&0.926685085623&{}:{}&0.136156269557&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.682466322424&{}:{}&1.023699483636&{}:{}&0.658766838788&. \end{alignedat} \]
1a (013)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1a}}&{}\approx{}&-0.070640817164&{}:{}&0.897603315436&{}:{}&0.173037501728&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.115085648128&{}:{}&0.934864950700&{}:{}&0.180220697428&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.263443026642&{}:{}&0.618128815849&{}:{}&0.645314210793&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.066243810640&{}:{}&0.841732392764&{}:{}&0.224511417877&. \end{alignedat} \]
1a (013)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1a}}&{}\approx{}&-0.018051002358&{}:{}&0.955505367660&{}:{}&0.062545634699&,\\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.067973133574&{}:{}&1.002360450786&{}:{}&0.065612682789&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.127399428194&{}:{}&0.685968091671&{}:{}&0.441431336523&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.016747305823&{}:{}&0.886495956863&{}:{}&0.130251348960&. \end{alignedat} \]
1a (013)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1a}}&{}\approx{}&-0.109724275369&{}:{}&0.854571932906&{}:{}&0.255152342463&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.082224400214&{}:{}&0.442728163724&{}:{}&0.639496236490&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.062841355180&{}:{}&0.926685085623&{}:{}&0.136156269557&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-0.682466322424&{}:{}&1.023699483636&{}:{}&0.658766838788&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.115085648128&{}:{}&0.934864950700&{}:{}&0.180220697428&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.263443026642&{}:{}&0.618128815849&{}:{}&0.645314210793&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.066243810640&{}:{}&0.841732392764&{}:{}&0.224511417877&, \end{alignedat} \]
1a (013)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1a}}&{}\approx{}&0.085333645441&{}:{}&1.138147853490&{}:{}&-0.223481498931&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.244331168209&{}:{}&-0.244331168209&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.617697946698&{}:{}&0.000000000000&{}:{}&1.617697946698&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.069746576075&{}:{}&0.930253423925&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1a}}&{}\approx{}&-0.176381553416&{}:{}&0.770039776868&{}:{}&0.406341776547&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.115085648128&{}:{}&0.934864950700&{}:{}&0.180220697428&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.263443026642&{}:{}&0.618128815849&{}:{}&0.645314210793&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.066243810640&{}:{}&0.841732392764&{}:{}&0.224511417877&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.244331168209&{}:{}&-0.244331168209&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.617697946698&{}:{}&0.000000000000&{}:{}&1.617697946698&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.069746576075&{}:{}&0.930253423925&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Hiroyasu Kamo