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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\(\mathbf{7b}\) \((323)\)

Malfatti circles

7b (323)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&, \\ P_{\mathbf{7b}}&{}\approx{}&-0.001273090162&{}:{}&1.002433288233&{}:{}&-0.001160198070&, \\ P^-_{\mathbf{7b}}&{}\approx{}&0.006480409818&{}:{}&0.977901146842&{}:{}&0.015618443340&, \\ P^+_{\mathbf{7b}}&{}\approx{}&-0.009231848736&{}:{}&1.027614869600&{}:{}&-0.018383020864&, \\ Q_{\mathbf{7b}}&{}\approx{}&0.113159993522&{}:{}&0.848646950554&{}:{}&0.038193055924&, \\ I^\prime_{\mathbf{7b}}&{}\approx{}&-0.031287163781&{}:{}&1.075251319245&{}:{}&-0.043964155463&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{7b}}\)
\(P^-_{\mathbf{7b}}\)
\(P^+_{\mathbf{7b}}\)
\(Q_{\mathbf{7b}}\)
\(I^\prime_{\mathbf{7b}}\)
7b (323)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7b}}&{}\approx{}&4.900660823394&{}:{}&8.776486852636&{}:{}&-12.677147676029&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.008115250889&{}:{}&1.025698294480&{}:{}&-0.017583043592&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.397038500742&{}:{}&0.595557751114&{}:{}&0.801480749629&, \\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.015905403975&{}:{}&1.017082556843&{}:{}&-0.001177152868&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.060389879342&{}:{}&1.115424647156&{}:{}&-0.055034767814&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.001316270618&{}:{}&1.036433650440&{}:{}&-0.035117379822&, \\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.007848671742&{}:{}&0.992004970691&{}:{}&0.015843701051&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.192271477853&{}:{}&0.344334806739&{}:{}&0.463393715408&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.006697282034&{}:{}&1.010627408744&{}:{}&-0.017324690778&, \\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.956933544219&{}:{}&0.043066455781&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.747655854551&{}:{}&0.000000000000&{}:{}&0.252344145449&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.117653541825&{}:{}&0.882346458175&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7b}}}{B^\prime_{\mathbf{7b}}}{C^\prime_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^*_{\mathbf{7b}}}{B^*_{\mathbf{7b}}}{C^*_{\mathbf{7b}}}\)
7b (323)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7b}}}}&{}\approx{}&-9.751652058484&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.013525418148&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.661730834571&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&4.900660823394&{}:{}&8.776486852636&{}:{}&-12.677147676029&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.008115250889&{}:{}&1.025698294480&{}:{}&-0.017583043592&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.397038500742&{}:{}&0.595557751114&{}:{}&0.801480749629&. \end{alignedat} \]
7b (323)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7b}}&{}\approx{}&-0.001273090162&{}:{}&1.002433288233&{}:{}&-0.001160198070&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.015905403975&{}:{}&1.017082556843&{}:{}&-0.001177152868&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.060389879342&{}:{}&1.115424647156&{}:{}&-0.055034767814&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.001316270618&{}:{}&1.036433650440&{}:{}&-0.035117379822&. \end{alignedat} \]
7b (323)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7b}}&{}\approx{}&0.006480409818&{}:{}&0.977901146842&{}:{}&0.015618443340&,\\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.007848671742&{}:{}&0.992004970691&{}:{}&0.015843701051&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.192271477853&{}:{}&0.344334806739&{}:{}&0.463393715408&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.006697282034&{}:{}&1.010627408744&{}:{}&-0.017324690778&. \end{alignedat} \]
7b (323)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7b}}&{}\approx{}&-0.009231848736&{}:{}&1.027614869600&{}:{}&-0.018383020864&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&4.900660823394&{}:{}&8.776486852636&{}:{}&-12.677147676029&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.008115250889&{}:{}&1.025698294480&{}:{}&-0.017583043592&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.397038500742&{}:{}&0.595557751114&{}:{}&0.801480749629&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.015905403975&{}:{}&1.017082556843&{}:{}&-0.001177152868&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.060389879342&{}:{}&1.115424647156&{}:{}&-0.055034767814&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.001316270618&{}:{}&1.036433650440&{}:{}&-0.035117379822&, \end{alignedat} \]
7b (323)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7b}}&{}\approx{}&0.113159993522&{}:{}&0.848646950554&{}:{}&0.038193055924&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.956933544219&{}:{}&0.043066455781&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.747655854551&{}:{}&0.000000000000&{}:{}&0.252344145449&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.117653541825&{}:{}&0.882346458175&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7b}}&{}\approx{}&-0.031287163781&{}:{}&1.075251319245&{}:{}&-0.043964155463&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.015905403975&{}:{}&1.017082556843&{}:{}&-0.001177152868&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.060389879342&{}:{}&1.115424647156&{}:{}&-0.055034767814&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.001316270618&{}:{}&1.036433650440&{}:{}&-0.035117379822&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.956933544219&{}:{}&0.043066455781&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.747655854551&{}:{}&0.000000000000&{}:{}&0.252344145449&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.117653541825&{}:{}&0.882346458175&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Hiroyasu Kamo