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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\(\mathbf{4a}\) \((211)\)

Malfatti circles

4a (211)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&, \\ P_{\mathbf{4a}}&{}\approx{}&1.036390092568&{}:{}&-0.008336706989&{}:{}&-0.028053385579&, \\ P^-_{\mathbf{4a}}&{}\approx{}&0.688895567474&{}:{}&0.150817025874&{}:{}&0.160287406652&, \\ P^+_{\mathbf{4a}}&{}\approx{}&1.725269776856&{}:{}&-0.323846061326&{}:{}&-0.401423715530&, \\ Q_{\mathbf{4a}}&{}\approx{}&2.201993535020&{}:{}&-0.502117975910&{}:{}&-0.699875559110&, \\ I^\prime_{\mathbf{4a}}&{}\approx{}&1.540142455168&{}:{}&-0.181792871182&{}:{}&-0.358349583985&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{4a}}\)
\(P^-_{\mathbf{4a}}\)
\(P^+_{\mathbf{4a}}\)
\(Q_{\mathbf{4a}}\)
\(I^\prime_{\mathbf{4a}}\)
4a (211)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4a}}&{}\approx{}&1.965925826289&{}:{}&-0.448287736084&{}:{}&-0.517638090205&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.508507942876&{}:{}&2.508507942876&{}:{}&-3.017015885753&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.965925826289&{}:{}&-1.673032607476&{}:{}&1.707106781187&, \\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.177729466308&{}:{}&-0.040716535172&{}:{}&-0.137012931136&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.754600960884&{}:{}&-0.707106781187&{}:{}&-0.047494179697&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.898938528181&{}:{}&-0.015275034192&{}:{}&-0.883663493989&, \\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.225721727966&{}:{}&0.375354170427&{}:{}&0.398924101607&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.995377271606&{}:{}&-0.226974699488&{}:{}&0.231597427882&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.046440152556&{}:{}&0.229092766764&{}:{}&-0.275532919319&, \\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.417737667700&{}:{}&0.582262332300&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.465925826289&{}:{}&0.000000000000&{}:{}&-0.465925826289&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.295385137588&{}:{}&-0.295385137588&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4a}}}{B^\prime_{\mathbf{4a}}}{C^\prime_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^*_{\mathbf{4a}}}{B^*_{\mathbf{4a}}}{C^*_{\mathbf{4a}}}\)
4a (211)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4a}}}}&{}\approx{}&-0.707106781187&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4a}}}}&{}\approx{}&-4.121320343560&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4a}}}}&{}\approx{}&-2.638958433765&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&1.965925826289&{}:{}&-0.448287736084&{}:{}&-0.517638090205&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.508507942876&{}:{}&2.508507942876&{}:{}&-3.017015885753&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.965925826289&{}:{}&-1.673032607476&{}:{}&1.707106781187&. \end{alignedat} \]
4a (211)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4a}}&{}\approx{}&1.036390092568&{}:{}&-0.008336706989&{}:{}&-0.028053385579&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.177729466308&{}:{}&-0.040716535172&{}:{}&-0.137012931136&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.754600960884&{}:{}&-0.707106781187&{}:{}&-0.047494179697&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.898938528181&{}:{}&-0.015275034192&{}:{}&-0.883663493989&. \end{alignedat} \]
4a (211)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4a}}&{}\approx{}&0.688895567474&{}:{}&0.150817025874&{}:{}&0.160287406652&,\\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.225721727966&{}:{}&0.375354170427&{}:{}&0.398924101607&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.995377271606&{}:{}&-0.226974699488&{}:{}&0.231597427882&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.046440152556&{}:{}&0.229092766764&{}:{}&-0.275532919319&. \end{alignedat} \]
4a (211)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4a}}&{}\approx{}&1.725269776856&{}:{}&-0.323846061326&{}:{}&-0.401423715530&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&1.965925826289&{}:{}&-0.448287736084&{}:{}&-0.517638090205&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.508507942876&{}:{}&2.508507942876&{}:{}&-3.017015885753&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.965925826289&{}:{}&-1.673032607476&{}:{}&1.707106781187&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.177729466308&{}:{}&-0.040716535172&{}:{}&-0.137012931136&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.754600960884&{}:{}&-0.707106781187&{}:{}&-0.047494179697&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.898938528181&{}:{}&-0.015275034192&{}:{}&-0.883663493989&, \end{alignedat} \]
4a (211)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4a}}&{}\approx{}&2.201993535020&{}:{}&-0.502117975910&{}:{}&-0.699875559110&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.417737667700&{}:{}&0.582262332300&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.465925826289&{}:{}&0.000000000000&{}:{}&-0.465925826289&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.295385137588&{}:{}&-0.295385137588&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4a}}&{}\approx{}&1.540142455168&{}:{}&-0.181792871182&{}:{}&-0.358349583985&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.177729466308&{}:{}&-0.040716535172&{}:{}&-0.137012931136&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.754600960884&{}:{}&-0.707106781187&{}:{}&-0.047494179697&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.898938528181&{}:{}&-0.015275034192&{}:{}&-0.883663493989&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.417737667700&{}:{}&0.582262332300&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.465925826289&{}:{}&0.000000000000&{}:{}&-0.465925826289&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.295385137588&{}:{}&-0.295385137588&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Hiroyasu Kamo