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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\(\mathbf{0c}\) \((110)\)

Malfatti circles

0c (110)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&, \\ P_{\mathbf{c}}&{}\approx{}&1.152256660770&{}:{}&1.443910539004&{}:{}&-1.596167199774&, \\ P^-_{\mathbf{c}}&{}\approx{}&0.909064932799&{}:{}&1.041681689858&{}:{}&-0.950746622657&, \\ P^+_{\mathbf{c}}&{}\approx{}&1.344772292931&{}:{}&1.762323265507&{}:{}&-2.107095558438&, \\ Q_{\mathbf{c}}&{}\approx{}&0.846301882882&{}:{}&0.748963831077&{}:{}&-0.595265713959&, \\ I^\prime_{\mathbf{c}}&{}\approx{}&1.566018606833&{}:{}&2.147029822589&{}:{}&-2.713048429421&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{c}}\)
\(P^-_{\mathbf{c}}\)
\(P^+_{\mathbf{c}}\)
\(Q_{\mathbf{c}}\)
\(I^\prime_{\mathbf{c}}\)
0c (110)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{c}}&{}\approx{}&1.407583257728&{}:{}&0.917062329887&{}:{}&-1.324645587615&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.679566334881&{}:{}&1.792827390695&{}:{}&-1.472393725577&,\\C^\prime_{\mathbf{c}}&{}\approx{}&10.856414843638&{}:{}&16.284622265457&{}:{}&-26.141037109095&, \\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.279068322161&{}:{}&2.646516017306&{}:{}&-2.925584339466&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.911322729339&{}:{}&1.736343153291&{}:{}&-2.647665882630&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.062802832222&{}:{}&1.331814570986&{}:{}&-1.394617403208&, \\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.812806714508&{}:{}&2.144341275183&{}:{}&-1.957147989691&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.651079526095&{}:{}&1.075703926370&{}:{}&-1.726783452465&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.830010086176&{}:{}&0.951094116572&{}:{}&-0.781104202748&, \\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&4.872953846944&{}:{}&-3.872953846944&,\\B^*_{\mathbf{c}}&{}\approx{}&3.371234856368&{}:{}&0.000000000000&{}:{}&-2.371234856368&,\\C^*_{\mathbf{c}}&{}\approx{}&0.530508413411&{}:{}&0.469491586589&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{c}}}{B^\prime_{\mathbf{c}}}{C^\prime_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{c}}}{B^{\prime\prime}_{\mathbf{c}}}{C^{\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{c}}}{B^{\prime\prime\prime}_{\mathbf{c}}}{C^{\prime\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^*_{\mathbf{c}}}{B^*_{\mathbf{c}}}{C^*_{\mathbf{c}}}\)
0c (110)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{c}}}}&{}\approx{}&0.203791628864&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{c}}}}&{}\approx{}&0.226522111627&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{c}}}}&{}\approx{}&3.618804947879&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.407583257728&{}:{}&0.917062329887&{}:{}&-1.324645587615&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.679566334881&{}:{}&1.792827390695&{}:{}&-1.472393725577&,\\C^\prime_{\mathbf{c}}&{}\approx{}&10.856414843638&{}:{}&16.284622265457&{}:{}&-26.141037109095&. \end{alignedat} \]
0c (110)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{c}}&{}\approx{}&1.152256660770&{}:{}&1.443910539004&{}:{}&-1.596167199774&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.279068322161&{}:{}&2.646516017306&{}:{}&-2.925584339466&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.911322729339&{}:{}&1.736343153291&{}:{}&-2.647665882630&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.062802832222&{}:{}&1.331814570986&{}:{}&-1.394617403208&. \end{alignedat} \]
0c (110)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{c}}&{}\approx{}&0.909064932799&{}:{}&1.041681689858&{}:{}&-0.950746622657&,\\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.812806714508&{}:{}&2.144341275183&{}:{}&-1.957147989691&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.651079526095&{}:{}&1.075703926370&{}:{}&-1.726783452465&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.830010086176&{}:{}&0.951094116572&{}:{}&-0.781104202748&. \end{alignedat} \]
0c (110)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{c}}&{}\approx{}&1.344772292931&{}:{}&1.762323265507&{}:{}&-2.107095558438&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.407583257728&{}:{}&0.917062329887&{}:{}&-1.324645587615&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.679566334881&{}:{}&1.792827390695&{}:{}&-1.472393725577&,\\C^\prime_{\mathbf{c}}&{}\approx{}&10.856414843638&{}:{}&16.284622265457&{}:{}&-26.141037109095&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.279068322161&{}:{}&2.646516017306&{}:{}&-2.925584339466&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.911322729339&{}:{}&1.736343153291&{}:{}&-2.647665882630&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.062802832222&{}:{}&1.331814570986&{}:{}&-1.394617403208&, \end{alignedat} \]
0c (110)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{c}}&{}\approx{}&0.846301882882&{}:{}&0.748963831077&{}:{}&-0.595265713959&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&4.872953846944&{}:{}&-3.872953846944&,\\B^*_{\mathbf{c}}&{}\approx{}&3.371234856368&{}:{}&0.000000000000&{}:{}&-2.371234856368&,\\C^*_{\mathbf{c}}&{}\approx{}&0.530508413411&{}:{}&0.469491586589&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{c}}&{}\approx{}&1.566018606833&{}:{}&2.147029822589&{}:{}&-2.713048429421&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.279068322161&{}:{}&2.646516017306&{}:{}&-2.925584339466&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.911322729339&{}:{}&1.736343153291&{}:{}&-2.647665882630&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.062802832222&{}:{}&1.331814570986&{}:{}&-1.394617403208&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&4.872953846944&{}:{}&-3.872953846944&,\\B^*_{\mathbf{c}}&{}\approx{}&3.371234856368&{}:{}&0.000000000000&{}:{}&-2.371234856368&,\\C^*_{\mathbf{c}}&{}\approx{}&0.530508413411&{}:{}&0.469491586589&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Hiroyasu Kamo