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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\(\mathbf{6b}\) \((321)\)

Malfatti circles

6b (321)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&, \\ P_{\mathbf{6b}}&{}\approx{}&-0.019367170355&{}:{}&1.026668858764&{}:{}&-0.007301688408&, \\ P^-_{\mathbf{6b}}&{}\approx{}&0.011554817056&{}:{}&0.935105754440&{}:{}&0.053339428504&, \\ P^+_{\mathbf{6b}}&{}\approx{}&-0.052851933015&{}:{}&1.125820597489&{}:{}&-0.072968664474&, \\ Q_{\mathbf{6b}}&{}\approx{}&-0.458391662643&{}:{}&1.312697719368&{}:{}&0.145693943275&, \\ I^\prime_{\mathbf{6b}}&{}\approx{}&-0.129637951449&{}:{}&1.242208742807&{}:{}&-0.112570791359&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{6b}}\)
\(P^-_{\mathbf{6b}}\)
\(P^+_{\mathbf{6b}}\)
\(Q_{\mathbf{6b}}\)
\(I^\prime_{\mathbf{6b}}\)
6b (321)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6b}}&{}\approx{}&3.331951230074&{}:{}&15.073969712741&{}:{}&-17.405920942814&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.035735452894&{}:{}&1.107206358682&{}:{}&-0.071470905788&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.400099577495&{}:{}&0.692992796309&{}:{}&0.707106781187&, \\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.065610786686&{}:{}&1.073243716365&{}:{}&-0.007632929680&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.236067708731&{}:{}&1.325068467273&{}:{}&-0.089000758542&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.021963854722&{}:{}&1.164321129380&{}:{}&-0.142357274659&, \\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.031495327193&{}:{}&0.975832785450&{}:{}&0.055662541743&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.098616666258&{}:{}&0.446148379041&{}:{}&0.455234954701&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.013037156421&{}:{}&1.055068196363&{}:{}&-0.068105352783&, \\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.900099577495&{}:{}&0.099900422505&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.465925826289&{}:{}&0.000000000000&{}:{}&-0.465925826289&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.536566092485&{}:{}&1.536566092485&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6b}}}{B^\prime_{\mathbf{6b}}}{C^\prime_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^*_{\mathbf{6b}}}{B^*_{\mathbf{6b}}}{C^*_{\mathbf{6b}}}\)
6b (321)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6b}}}}&{}\approx{}&-11.034911701481&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.045310738638&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.507305936177&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&3.331951230074&{}:{}&15.073969712741&{}:{}&-17.405920942814&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.035735452894&{}:{}&1.107206358682&{}:{}&-0.071470905788&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.400099577495&{}:{}&0.692992796309&{}:{}&0.707106781187&. \end{alignedat} \]
6b (321)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6b}}&{}\approx{}&-0.019367170355&{}:{}&1.026668858764&{}:{}&-0.007301688408&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.065610786686&{}:{}&1.073243716365&{}:{}&-0.007632929680&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.236067708731&{}:{}&1.325068467273&{}:{}&-0.089000758542&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.021963854722&{}:{}&1.164321129380&{}:{}&-0.142357274659&. \end{alignedat} \]
6b (321)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6b}}&{}\approx{}&0.011554817056&{}:{}&0.935105754440&{}:{}&0.053339428504&,\\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.031495327193&{}:{}&0.975832785450&{}:{}&0.055662541743&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.098616666258&{}:{}&0.446148379041&{}:{}&0.455234954701&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.013037156421&{}:{}&1.055068196363&{}:{}&-0.068105352783&. \end{alignedat} \]
6b (321)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6b}}&{}\approx{}&-0.052851933015&{}:{}&1.125820597489&{}:{}&-0.072968664474&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&3.331951230074&{}:{}&15.073969712741&{}:{}&-17.405920942814&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.035735452894&{}:{}&1.107206358682&{}:{}&-0.071470905788&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.400099577495&{}:{}&0.692992796309&{}:{}&0.707106781187&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.065610786686&{}:{}&1.073243716365&{}:{}&-0.007632929680&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.236067708731&{}:{}&1.325068467273&{}:{}&-0.089000758542&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.021963854722&{}:{}&1.164321129380&{}:{}&-0.142357274659&, \end{alignedat} \]
6b (321)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6b}}&{}\approx{}&-0.458391662643&{}:{}&1.312697719368&{}:{}&0.145693943275&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.900099577495&{}:{}&0.099900422505&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.465925826289&{}:{}&0.000000000000&{}:{}&-0.465925826289&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.536566092485&{}:{}&1.536566092485&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6b}}&{}\approx{}&-0.129637951449&{}:{}&1.242208742807&{}:{}&-0.112570791359&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.065610786686&{}:{}&1.073243716365&{}:{}&-0.007632929680&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.236067708731&{}:{}&1.325068467273&{}:{}&-0.089000758542&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.021963854722&{}:{}&1.164321129380&{}:{}&-0.142357274659&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.900099577495&{}:{}&0.099900422505&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.465925826289&{}:{}&0.000000000000&{}:{}&-0.465925826289&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.536566092485&{}:{}&1.536566092485&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Hiroyasu Kamo