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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\(\mathbf{3a}\) \((033)\)

Malfatti circles

3a (033)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&, \\ P_{\mathbf{3a}}&{}\approx{}&-0.008629550816&{}:{}&0.231069734436&{}:{}&0.777559816380&, \\ P^-_{\mathbf{3a}}&{}\approx{}&0.058559649996&{}:{}&0.155324995258&{}:{}&0.786115354746&, \\ P^+_{\mathbf{3a}}&{}\approx{}&-0.057459154054&{}:{}&0.286117056278&{}:{}&0.771342097776&, \\ Q_{\mathbf{3a}}&{}\approx{}&0.030774907897&{}:{}&0.404881829452&{}:{}&0.564343262651&, \\ I^\prime_{\mathbf{3a}}&{}\approx{}&-0.051990324626&{}:{}&0.354062784253&{}:{}&0.697927540373&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{3a}}\)
\(P^-_{\mathbf{3a}}\)
\(P^+_{\mathbf{3a}}\)
\(Q_{\mathbf{3a}}\)
\(I^\prime_{\mathbf{3a}}\)
3a (033)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3a}}&{}\approx{}&0.185686860132&{}:{}&0.377924043441&{}:{}&0.436389096428&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.307007203691&{}:{}&0.692992796309&{}:{}&0.614014407382&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.082262332300&{}:{}&0.142482539092&{}:{}&0.939779793208&, \\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.129756511997&{}:{}&0.258819045103&{}:{}&0.870937466894&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.008900306209&{}:{}&0.206944258973&{}:{}&0.801956047237&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.019002401230&{}:{}&0.508819045103&{}:{}&0.510183356128&, \\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.078855436697&{}:{}&0.177996636312&{}:{}&0.900858800384&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.060755344153&{}:{}&0.123653904788&{}:{}&0.815590751059&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.166595639717&{}:{}&0.441882199621&{}:{}&0.391522160661&, \\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.417737667700&{}:{}&0.582262332300&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.051712263916&{}:{}&0.000000000000&{}:{}&0.948287736084&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.070640266196&{}:{}&0.929359733804&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3a}}}{B^\prime_{\mathbf{3a}}}{C^\prime_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^*_{\mathbf{3a}}}{B^*_{\mathbf{3a}}}{C^*_{\mathbf{3a}}}\)
3a (033)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3a}}}}&{}\approx{}&0.596118591655&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3a}}}}&{}\approx{}&0.838759278774&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3a}}}}&{}\approx{}&0.224744871392&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.185686860132&{}:{}&0.377924043441&{}:{}&0.436389096428&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.307007203691&{}:{}&0.692992796309&{}:{}&0.614014407382&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.082262332300&{}:{}&0.142482539092&{}:{}&0.939779793208&. \end{alignedat} \]
3a (033)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3a}}&{}\approx{}&-0.008629550816&{}:{}&0.231069734436&{}:{}&0.777559816380&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.129756511997&{}:{}&0.258819045103&{}:{}&0.870937466894&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.008900306209&{}:{}&0.206944258973&{}:{}&0.801956047237&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.019002401230&{}:{}&0.508819045103&{}:{}&0.510183356128&. \end{alignedat} \]
3a (033)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3a}}&{}\approx{}&0.058559649996&{}:{}&0.155324995258&{}:{}&0.786115354746&,\\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.078855436697&{}:{}&0.177996636312&{}:{}&0.900858800384&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.060755344153&{}:{}&0.123653904788&{}:{}&0.815590751059&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.166595639717&{}:{}&0.441882199621&{}:{}&0.391522160661&. \end{alignedat} \]
3a (033)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3a}}&{}\approx{}&-0.057459154054&{}:{}&0.286117056278&{}:{}&0.771342097776&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.185686860132&{}:{}&0.377924043441&{}:{}&0.436389096428&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.307007203691&{}:{}&0.692992796309&{}:{}&0.614014407382&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.082262332300&{}:{}&0.142482539092&{}:{}&0.939779793208&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.129756511997&{}:{}&0.258819045103&{}:{}&0.870937466894&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.008900306209&{}:{}&0.206944258973&{}:{}&0.801956047237&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.019002401230&{}:{}&0.508819045103&{}:{}&0.510183356128&, \end{alignedat} \]
3a (033)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3a}}&{}\approx{}&0.030774907897&{}:{}&0.404881829452&{}:{}&0.564343262651&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.417737667700&{}:{}&0.582262332300&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.051712263916&{}:{}&0.000000000000&{}:{}&0.948287736084&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.070640266196&{}:{}&0.929359733804&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3a}}&{}\approx{}&-0.051990324626&{}:{}&0.354062784253&{}:{}&0.697927540373&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.129756511997&{}:{}&0.258819045103&{}:{}&0.870937466894&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.008900306209&{}:{}&0.206944258973&{}:{}&0.801956047237&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.019002401230&{}:{}&0.508819045103&{}:{}&0.510183356128&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.417737667700&{}:{}&0.582262332300&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.051712263916&{}:{}&0.000000000000&{}:{}&0.948287736084&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.070640266196&{}:{}&0.929359733804&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Hiroyasu Kamo