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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\(\mathbf{1c}\) \((112)\)

Malfatti circles

1c (112)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&, \\ P_{\mathbf{1c}}&{}\approx{}&-0.382159745195&{}:{}&-0.780717375460&{}:{}&2.162877120655&, \\ P^-_{\mathbf{1c}}&{}\approx{}&0.062674292678&{}:{}&-0.086178084012&{}:{}&1.023503791335&, \\ P^+_{\mathbf{1c}}&{}\approx{}&-0.985772505320&{}:{}&-1.723164991949&{}:{}&3.708937497269&, \\ Q_{\mathbf{1c}}&{}\approx{}&1.411945916145&{}:{}&1.595447520668&{}:{}&-2.007393436813&, \\ I^\prime_{\mathbf{1c}}&{}\approx{}&-1.331109313533&{}:{}&-2.330149601974&{}:{}&4.661258915508&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{1c}}\)
\(P^-_{\mathbf{1c}}\)
\(P^+_{\mathbf{1c}}\)
\(Q_{\mathbf{1c}}\)
\(I^\prime_{\mathbf{1c}}\)
1c (112)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1c}}&{}\approx{}&0.192416742272&{}:{}&-1.817062329887&{}:{}&2.624645587615&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.054566334881&{}:{}&-0.230327390695&{}:{}&2.284893725577&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-10.642129129352&{}:{}&-15.963193694029&{}:{}&27.605322823381&, \\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.918974661791&{}:{}&-1.648792221674&{}:{}&4.567766883465&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.914825197230&{}:{}&-3.262733437258&{}:{}&5.177558634487&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.474221901905&{}:{}&-0.968791933992&{}:{}&2.443013835897&, \\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.726978457596&{}:{}&-0.158779059876&{}:{}&1.885757517473&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.126788496599&{}:{}&-1.197310578655&{}:{}&2.070522082056&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.075384040660&{}:{}&-0.103654176404&{}:{}&1.028270135744&, \\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-3.872953846944&{}:{}&4.872953846944&,\\B^*_{\mathbf{1c}}&{}\approx{}&-2.371234856368&{}:{}&0.000000000000&{}:{}&3.371234856368&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.469491586589&{}:{}&0.530508413411&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1c}}}{B^\prime_{\mathbf{1c}}}{C^\prime_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^*_{\mathbf{1c}}}{B^*_{\mathbf{1c}}}{C^*_{\mathbf{1c}}}\)
1c (112)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.403791628864&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.351522111627&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1c}}}}&{}\approx{}&-3.547376376451&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.192416742272&{}:{}&-1.817062329887&{}:{}&2.624645587615&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.054566334881&{}:{}&-0.230327390695&{}:{}&2.284893725577&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-10.642129129352&{}:{}&-15.963193694029&{}:{}&27.605322823381&. \end{alignedat} \]
1c (112)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1c}}&{}\approx{}&-0.382159745195&{}:{}&-0.780717375460&{}:{}&2.162877120655&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.918974661791&{}:{}&-1.648792221674&{}:{}&4.567766883465&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.914825197230&{}:{}&-3.262733437258&{}:{}&5.177558634487&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.474221901905&{}:{}&-0.968791933992&{}:{}&2.443013835897&. \end{alignedat} \]
1c (112)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1c}}&{}\approx{}&0.062674292678&{}:{}&-0.086178084012&{}:{}&1.023503791335&,\\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.726978457596&{}:{}&-0.158779059876&{}:{}&1.885757517473&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.126788496599&{}:{}&-1.197310578655&{}:{}&2.070522082056&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.075384040660&{}:{}&-0.103654176404&{}:{}&1.028270135744&. \end{alignedat} \]
1c (112)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1c}}&{}\approx{}&-0.985772505320&{}:{}&-1.723164991949&{}:{}&3.708937497269&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.192416742272&{}:{}&-1.817062329887&{}:{}&2.624645587615&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.054566334881&{}:{}&-0.230327390695&{}:{}&2.284893725577&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-10.642129129352&{}:{}&-15.963193694029&{}:{}&27.605322823381&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.918974661791&{}:{}&-1.648792221674&{}:{}&4.567766883465&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.914825197230&{}:{}&-3.262733437258&{}:{}&5.177558634487&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.474221901905&{}:{}&-0.968791933992&{}:{}&2.443013835897&, \end{alignedat} \]
1c (112)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1c}}&{}\approx{}&1.411945916145&{}:{}&1.595447520668&{}:{}&-2.007393436813&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-3.872953846944&{}:{}&4.872953846944&,\\B^*_{\mathbf{1c}}&{}\approx{}&-2.371234856368&{}:{}&0.000000000000&{}:{}&3.371234856368&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.469491586589&{}:{}&0.530508413411&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1c}}&{}\approx{}&-1.331109313533&{}:{}&-2.330149601974&{}:{}&4.661258915508&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.918974661791&{}:{}&-1.648792221674&{}:{}&4.567766883465&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.914825197230&{}:{}&-3.262733437258&{}:{}&5.177558634487&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.474221901905&{}:{}&-0.968791933992&{}:{}&2.443013835897&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-3.872953846944&{}:{}&4.872953846944&,\\B^*_{\mathbf{1c}}&{}\approx{}&-2.371234856368&{}:{}&0.000000000000&{}:{}&3.371234856368&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.469491586589&{}:{}&0.530508413411&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Hiroyasu Kamo