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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\(\mathbf{4b}\) \((301)\)

Malfatti circles

4b (301)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&, \\ P_{\mathbf{4b}}&{}\approx{}&0.006137522517&{}:{}&-0.003535258166&{}:{}&0.997397735649&, \\ P^-_{\mathbf{4b}}&{}\approx{}&-0.007685159222&{}:{}&0.017330762649&{}:{}&0.990354396573&, \\ P^+_{\mathbf{4b}}&{}\approx{}&0.019345355921&{}:{}&-0.023473135052&{}:{}&1.004127779130&, \\ Q_{\mathbf{4b}}&{}\approx{}&-0.269549689511&{}:{}&0.054675005565&{}:{}&1.214874683946&, \\ I^\prime_{\mathbf{4b}}&{}\approx{}&0.053093111173&{}:{}&-0.049351211954&{}:{}&0.996258100782&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{4b}}\)
\(P^-_{\mathbf{4b}}\)
\(P^+_{\mathbf{4b}}\)
\(Q_{\mathbf{4b}}\)
\(I^\prime_{\mathbf{4b}}\)
4b (301)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4b}}&{}\approx{}&-4.728748213577&{}:{}&-12.889683480548&{}:{}&18.618431694126&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.226879143281&{}:{}&0.281549379609&{}:{}&0.491571477110&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.014201689055&{}:{}&-0.021302533582&{}:{}&1.007100844527&, \\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.026730180203&{}:{}&-0.003462008232&{}:{}&0.976731828029&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.006376039684&{}:{}&-0.042534770762&{}:{}&1.036158731078&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.099404279045&{}:{}&-0.057257596738&{}:{}&0.957853317693&, \\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.013669582095&{}:{}&0.016963491235&{}:{}&0.969366926669&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.007991049150&{}:{}&-0.021782105870&{}:{}&1.029773155020&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.192589899859&{}:{}&0.434308482969&{}:{}&0.758281416890&, \\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.043066455781&{}:{}&0.956933544219&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.285139704438&{}:{}&0.000000000000&{}:{}&1.285139704438&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.254450661943&{}:{}&-0.254450661943&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4b}}}{B^\prime_{\mathbf{4b}}}{C^\prime_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^*_{\mathbf{4b}}}{B^*_{\mathbf{4b}}}{C^*_{\mathbf{4b}}}\)
4b (301)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4b}}}}&{}\approx{}&14.321870533943&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4b}}}}&{}\approx{}&0.378131905469&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4b}}}}&{}\approx{}&0.023669481758&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-4.728748213577&{}:{}&-12.889683480548&{}:{}&18.618431694126&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.226879143281&{}:{}&0.281549379609&{}:{}&0.491571477110&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.014201689055&{}:{}&-0.021302533582&{}:{}&1.007100844527&. \end{alignedat} \]
4b (301)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4b}}&{}\approx{}&0.006137522517&{}:{}&-0.003535258166&{}:{}&0.997397735649&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.026730180203&{}:{}&-0.003462008232&{}:{}&0.976731828029&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.006376039684&{}:{}&-0.042534770762&{}:{}&1.036158731078&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.099404279045&{}:{}&-0.057257596738&{}:{}&0.957853317693&. \end{alignedat} \]
4b (301)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4b}}&{}\approx{}&-0.007685159222&{}:{}&0.017330762649&{}:{}&0.990354396573&,\\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.013669582095&{}:{}&0.016963491235&{}:{}&0.969366926669&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.007991049150&{}:{}&-0.021782105870&{}:{}&1.029773155020&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.192589899859&{}:{}&0.434308482969&{}:{}&0.758281416890&. \end{alignedat} \]
4b (301)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4b}}&{}\approx{}&0.019345355921&{}:{}&-0.023473135052&{}:{}&1.004127779130&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-4.728748213577&{}:{}&-12.889683480548&{}:{}&18.618431694126&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.226879143281&{}:{}&0.281549379609&{}:{}&0.491571477110&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.014201689055&{}:{}&-0.021302533582&{}:{}&1.007100844527&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.026730180203&{}:{}&-0.003462008232&{}:{}&0.976731828029&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.006376039684&{}:{}&-0.042534770762&{}:{}&1.036158731078&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.099404279045&{}:{}&-0.057257596738&{}:{}&0.957853317693&, \end{alignedat} \]
4b (301)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4b}}&{}\approx{}&-0.269549689511&{}:{}&0.054675005565&{}:{}&1.214874683946&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.043066455781&{}:{}&0.956933544219&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.285139704438&{}:{}&0.000000000000&{}:{}&1.285139704438&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.254450661943&{}:{}&-0.254450661943&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4b}}&{}\approx{}&0.053093111173&{}:{}&-0.049351211954&{}:{}&0.996258100782&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.026730180203&{}:{}&-0.003462008232&{}:{}&0.976731828029&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.006376039684&{}:{}&-0.042534770762&{}:{}&1.036158731078&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.099404279045&{}:{}&-0.057257596738&{}:{}&0.957853317693&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.043066455781&{}:{}&0.956933544219&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.285139704438&{}:{}&0.000000000000&{}:{}&1.285139704438&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.254450661943&{}:{}&-0.254450661943&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Hiroyasu Kamo