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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\(\mathbf{0b}\) \((101)\)

Malfatti circles

0b (101)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&, \\ P_{\mathbf{b}}&{}\approx{}&0.769653227915&{}:{}&-0.606662609039&{}:{}&0.837009381123&, \\ P^-_{\mathbf{b}}&{}\approx{}&0.765109255820&{}:{}&-0.425265265030&{}:{}&0.660156009210&, \\ P^+_{\mathbf{b}}&{}\approx{}&0.772728129079&{}:{}&-0.729414001978&{}:{}&0.956685872899&, \\ Q_{\mathbf{b}}&{}\approx{}&0.839901758402&{}:{}&-0.293292170627&{}:{}&0.453390412225&, \\ I^\prime_{\mathbf{b}}&{}\approx{}&0.765486747582&{}:{}&-0.894425261855&{}:{}&1.128938514273&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{b}}\)
\(P^-_{\mathbf{b}}\)
\(P^+_{\mathbf{b}}\)
\(Q_{\mathbf{b}}\)
\(I^\prime_{\mathbf{b}}\)
0b (101)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{b}}&{}\approx{}&0.938104379956&{}:{}&-0.400099577495&{}:{}&0.461995197539&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.346352670420&{}:{}&-3.039058011260&{}:{}&2.692705340840&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.360754110767&{}:{}&-0.624844448887&{}:{}&1.264090338120&, \\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.569035593729&{}:{}&-1.135027805015&{}:{}&1.565992211286&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.712728561547&{}:{}&-0.487831517751&{}:{}&0.775102956204&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.997890957891&{}:{}&-0.786566092485&{}:{}&0.788675134595&, \\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.445083498629&{}:{}&-1.004665866461&{}:{}&1.559582367831&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.696219866445&{}:{}&-0.296936332843&{}:{}&0.600716466399&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&1.067629565656&{}:{}&-0.593412988719&{}:{}&0.525783423063&, \\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.831951230074&{}:{}&2.831951230074&,\\B^*_{\mathbf{b}}&{}\approx{}&0.649429245361&{}:{}&0.000000000000&{}:{}&0.350570754639&,\\C^*_{\mathbf{b}}&{}\approx{}&1.536566092485&{}:{}&-0.536566092485&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{b}}}{B^\prime_{\mathbf{b}}}{C^\prime_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{b}}}{B^{\prime\prime}_{\mathbf{b}}}{C^{\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{b}}}{B^{\prime\prime\prime}_{\mathbf{b}}}{C^{\prime\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^*_{\mathbf{b}}}{B^*_{\mathbf{b}}}{C^*_{\mathbf{b}}}\)
0b (101)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{b}}}}&{}\approx{}&0.292893218813&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{b}}}}&{}\approx{}&1.707106781187&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{b}}}}&{}\approx{}&0.457417883413&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.938104379956&{}:{}&-0.400099577495&{}:{}&0.461995197539&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.346352670420&{}:{}&-3.039058011260&{}:{}&2.692705340840&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.360754110767&{}:{}&-0.624844448887&{}:{}&1.264090338120&. \end{alignedat} \]
0b (101)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{b}}&{}\approx{}&0.769653227915&{}:{}&-0.606662609039&{}:{}&0.837009381123&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.569035593729&{}:{}&-1.135027805015&{}:{}&1.565992211286&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.712728561547&{}:{}&-0.487831517751&{}:{}&0.775102956204&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.997890957891&{}:{}&-0.786566092485&{}:{}&0.788675134595&. \end{alignedat} \]
0b (101)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{b}}&{}\approx{}&0.765109255820&{}:{}&-0.425265265030&{}:{}&0.660156009210&,\\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.445083498629&{}:{}&-1.004665866461&{}:{}&1.559582367831&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.696219866445&{}:{}&-0.296936332843&{}:{}&0.600716466399&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&1.067629565656&{}:{}&-0.593412988719&{}:{}&0.525783423063&. \end{alignedat} \]
0b (101)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{b}}&{}\approx{}&0.772728129079&{}:{}&-0.729414001978&{}:{}&0.956685872899&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.938104379956&{}:{}&-0.400099577495&{}:{}&0.461995197539&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.346352670420&{}:{}&-3.039058011260&{}:{}&2.692705340840&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.360754110767&{}:{}&-0.624844448887&{}:{}&1.264090338120&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.569035593729&{}:{}&-1.135027805015&{}:{}&1.565992211286&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.712728561547&{}:{}&-0.487831517751&{}:{}&0.775102956204&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.997890957891&{}:{}&-0.786566092485&{}:{}&0.788675134595&, \end{alignedat} \]
0b (101)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{b}}&{}\approx{}&0.839901758402&{}:{}&-0.293292170627&{}:{}&0.453390412225&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.831951230074&{}:{}&2.831951230074&,\\B^*_{\mathbf{b}}&{}\approx{}&0.649429245361&{}:{}&0.000000000000&{}:{}&0.350570754639&,\\C^*_{\mathbf{b}}&{}\approx{}&1.536566092485&{}:{}&-0.536566092485&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{b}}&{}\approx{}&0.765486747582&{}:{}&-0.894425261855&{}:{}&1.128938514273&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.569035593729&{}:{}&-1.135027805015&{}:{}&1.565992211286&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.712728561547&{}:{}&-0.487831517751&{}:{}&0.775102956204&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.997890957891&{}:{}&-0.786566092485&{}:{}&0.788675134595&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.831951230074&{}:{}&2.831951230074&,\\B^*_{\mathbf{b}}&{}\approx{}&0.649429245361&{}:{}&0.000000000000&{}:{}&0.350570754639&,\\C^*_{\mathbf{b}}&{}\approx{}&1.536566092485&{}:{}&-0.536566092485&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Hiroyasu Kamo