Derousseau's Generalization of the Malfatti circles

Half of an Equilateral Triangle

\(A=30\degree\), \(B=60\degree\), \(C=90\degree\).


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\(\mathbf{7b}\) \((323)\)

Malfatti circles

7b (323)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&, \\ P_{\mathbf{7b}}&{}\approx{}&-0.006630211516&{}:{}&1.013840665571&{}:{}&-0.007210454055&, \\ P^-_{\mathbf{7b}}&{}\approx{}&0.011353533503&{}:{}&0.960026234699&{}:{}&0.028620231798&, \\ P^+_{\mathbf{7b}}&{}\approx{}&-0.025465791018&{}:{}&1.070204119900&{}:{}&-0.044738328882&, \\ Q_{\mathbf{7b}}&{}\approx{}&0.156164184623&{}:{}&0.759536260897&{}:{}&0.084299554481&, \\ I^\prime_{\mathbf{7b}}&{}\approx{}&-0.072466550216&{}:{}&1.179340091521&{}:{}&-0.106873541305&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{7b}}\)
\(P^-_{\mathbf{7b}}\)
\(P^+_{\mathbf{7b}}\)
\(Q_{\mathbf{7b}}\)
\(I^\prime_{\mathbf{7b}}\)
7b (323)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7b}}&{}\approx{}&1.726779514551&{}:{}&4.697976633858&{}:{}&-5.424756148409&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.019672733364&{}:{}&1.059018200093&{}:{}&-0.039345466729&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.427921023828&{}:{}&0.741180954897&{}:{}&0.686740068931&, \\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.037616146640&{}:{}&1.045048551774&{}:{}&-0.007432405134&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.134305151059&{}:{}&1.280364008856&{}:{}&-0.146058857797&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.007097019431&{}:{}&1.085221321497&{}:{}&-0.078124302066&, \\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.018369991613&{}:{}&0.988889296335&{}:{}&0.029480695278&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.160218419584&{}:{}&0.435899537363&{}:{}&0.403882043053&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.012133575236&{}:{}&1.025984601583&{}:{}&-0.038118176819&, \\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.900099577495&{}:{}&0.099900422505&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.649429245361&{}:{}&0.000000000000&{}:{}&0.350570754639&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.170540688701&{}:{}&0.829459311299&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7b}}}{B^\prime_{\mathbf{7b}}}{C^\prime_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^*_{\mathbf{7b}}}{B^*_{\mathbf{7b}}}{C^*_{\mathbf{7b}}}\)
7b (323)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7b}}}}&{}\approx{}&-3.439157588755&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.024944026382&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.542582116587&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&1.726779514551&{}:{}&4.697976633858&{}:{}&-5.424756148409&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.019672733364&{}:{}&1.059018200093&{}:{}&-0.039345466729&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.427921023828&{}:{}&0.741180954897&{}:{}&0.686740068931&. \end{alignedat} \]
7b (323)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7b}}&{}\approx{}&-0.006630211516&{}:{}&1.013840665571&{}:{}&-0.007210454055&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.037616146640&{}:{}&1.045048551774&{}:{}&-0.007432405134&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.134305151059&{}:{}&1.280364008856&{}:{}&-0.146058857797&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.007097019431&{}:{}&1.085221321497&{}:{}&-0.078124302066&. \end{alignedat} \]
7b (323)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7b}}&{}\approx{}&0.011353533503&{}:{}&0.960026234699&{}:{}&0.028620231798&,\\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.018369991613&{}:{}&0.988889296335&{}:{}&0.029480695278&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.160218419584&{}:{}&0.435899537363&{}:{}&0.403882043053&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.012133575236&{}:{}&1.025984601583&{}:{}&-0.038118176819&. \end{alignedat} \]
7b (323)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7b}}&{}\approx{}&-0.025465791018&{}:{}&1.070204119900&{}:{}&-0.044738328882&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&1.726779514551&{}:{}&4.697976633858&{}:{}&-5.424756148409&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.019672733364&{}:{}&1.059018200093&{}:{}&-0.039345466729&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.427921023828&{}:{}&0.741180954897&{}:{}&0.686740068931&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.037616146640&{}:{}&1.045048551774&{}:{}&-0.007432405134&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.134305151059&{}:{}&1.280364008856&{}:{}&-0.146058857797&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.007097019431&{}:{}&1.085221321497&{}:{}&-0.078124302066&, \end{alignedat} \]
7b (323)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7b}}&{}\approx{}&0.156164184623&{}:{}&0.759536260897&{}:{}&0.084299554481&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.900099577495&{}:{}&0.099900422505&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.649429245361&{}:{}&0.000000000000&{}:{}&0.350570754639&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.170540688701&{}:{}&0.829459311299&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7b}}&{}\approx{}&-0.072466550216&{}:{}&1.179340091521&{}:{}&-0.106873541305&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.037616146640&{}:{}&1.045048551774&{}:{}&-0.007432405134&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.134305151059&{}:{}&1.280364008856&{}:{}&-0.146058857797&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.007097019431&{}:{}&1.085221321497&{}:{}&-0.078124302066&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.900099577495&{}:{}&0.099900422505&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.649429245361&{}:{}&0.000000000000&{}:{}&0.350570754639&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.170540688701&{}:{}&0.829459311299&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Hiroyasu Kamo