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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\(\mathbf{0b}\) \((101)\)

Malfatti circles

0b (101)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&, \\ P_{\mathbf{b}}&{}\approx{}&0.748885850337&{}:{}&-0.431363764551&{}:{}&0.682477914214&, \\ P^-_{\mathbf{b}}&{}\approx{}&0.787949887418&{}:{}&-0.308404979429&{}:{}&0.520455092011&, \\ P^+_{\mathbf{b}}&{}\approx{}&0.723265913625&{}:{}&-0.512005616453&{}:{}&0.788739702828&, \\ Q_{\mathbf{b}}&{}\approx{}&0.881309275532&{}:{}&-0.178763290848&{}:{}&0.297454015316&, \\ I^\prime_{\mathbf{b}}&{}\approx{}&0.677662908379&{}:{}&-0.629902544536&{}:{}&0.952239636157&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{b}}\)
\(P^-_{\mathbf{b}}\)
\(P^+_{\mathbf{b}}\)
\(Q_{\mathbf{b}}\)
\(I^\prime_{\mathbf{b}}\)
0b (101)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{b}}&{}\approx{}&0.882461329374&{}:{}&-0.264462008909&{}:{}&0.382000679535&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.828080165740&{}:{}&-1.622253858178&{}:{}&1.794173692438&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.356373904386&{}:{}&-0.534560856579&{}:{}&1.178186952193&, \\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.498299037933&{}:{}&-0.861821669413&{}:{}&1.363522631480&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.708335977919&{}:{}&-0.353859805823&{}:{}&0.645523827905&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.872915384100&{}:{}&-0.502805689345&{}:{}&0.629890305246&, \\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.426078114356&{}:{}&-0.834710084236&{}:{}&1.408631969880&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.734844990160&{}:{}&-0.220223341086&{}:{}&0.485378350926&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.960172837354&{}:{}&-0.375813346611&{}:{}&0.415640509257&, \\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.506126882702&{}:{}&2.506126882702&,\\B^*_{\mathbf{b}}&{}\approx{}&0.747655854551&{}:{}&0.000000000000&{}:{}&0.252344145449&,\\C^*_{\mathbf{b}}&{}\approx{}&1.254450661943&{}:{}&-0.254450661943&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{b}}}{B^\prime_{\mathbf{b}}}{C^\prime_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{b}}}{B^{\prime\prime}_{\mathbf{b}}}{C^{\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{b}}}{B^{\prime\prime\prime}_{\mathbf{b}}}{C^{\prime\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^*_{\mathbf{b}}}{B^*_{\mathbf{b}}}{C^*_{\mathbf{b}}}\)
0b (101)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{b}}}}&{}\approx{}&0.293846676565&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{b}}}}&{}\approx{}&1.380133609567&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{b}}}}&{}\approx{}&0.593956507310&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.882461329374&{}:{}&-0.264462008909&{}:{}&0.382000679535&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.828080165740&{}:{}&-1.622253858178&{}:{}&1.794173692438&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.356373904386&{}:{}&-0.534560856579&{}:{}&1.178186952193&. \end{alignedat} \]
0b (101)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{b}}&{}\approx{}&0.748885850337&{}:{}&-0.431363764551&{}:{}&0.682477914214&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.498299037933&{}:{}&-0.861821669413&{}:{}&1.363522631480&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.708335977919&{}:{}&-0.353859805823&{}:{}&0.645523827905&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.872915384100&{}:{}&-0.502805689345&{}:{}&0.629890305246&. \end{alignedat} \]
0b (101)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{b}}&{}\approx{}&0.787949887418&{}:{}&-0.308404979429&{}:{}&0.520455092011&,\\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.426078114356&{}:{}&-0.834710084236&{}:{}&1.408631969880&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.734844990160&{}:{}&-0.220223341086&{}:{}&0.485378350926&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.960172837354&{}:{}&-0.375813346611&{}:{}&0.415640509257&. \end{alignedat} \]
0b (101)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{b}}&{}\approx{}&0.723265913625&{}:{}&-0.512005616453&{}:{}&0.788739702828&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.882461329374&{}:{}&-0.264462008909&{}:{}&0.382000679535&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.828080165740&{}:{}&-1.622253858178&{}:{}&1.794173692438&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.356373904386&{}:{}&-0.534560856579&{}:{}&1.178186952193&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.498299037933&{}:{}&-0.861821669413&{}:{}&1.363522631480&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.708335977919&{}:{}&-0.353859805823&{}:{}&0.645523827905&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.872915384100&{}:{}&-0.502805689345&{}:{}&0.629890305246&, \end{alignedat} \]
0b (101)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{b}}&{}\approx{}&0.881309275532&{}:{}&-0.178763290848&{}:{}&0.297454015316&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.506126882702&{}:{}&2.506126882702&,\\B^*_{\mathbf{b}}&{}\approx{}&0.747655854551&{}:{}&0.000000000000&{}:{}&0.252344145449&,\\C^*_{\mathbf{b}}&{}\approx{}&1.254450661943&{}:{}&-0.254450661943&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{b}}&{}\approx{}&0.677662908379&{}:{}&-0.629902544536&{}:{}&0.952239636157&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.498299037933&{}:{}&-0.861821669413&{}:{}&1.363522631480&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.708335977919&{}:{}&-0.353859805823&{}:{}&0.645523827905&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.872915384100&{}:{}&-0.502805689345&{}:{}&0.629890305246&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.506126882702&{}:{}&2.506126882702&,\\B^*_{\mathbf{b}}&{}\approx{}&0.747655854551&{}:{}&0.000000000000&{}:{}&0.252344145449&,\\C^*_{\mathbf{b}}&{}\approx{}&1.254450661943&{}:{}&-0.254450661943&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Hiroyasu Kamo