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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[Lob & Richmond]

\(\mathbf{0a}\) \((011)\)

Malfatti circles

0a (011)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&, \\ P_{\mathbf{a}}&{}\approx{}&-0.216094851447&{}:{}&0.642919776480&{}:{}&0.573175074967&, \\ P^-_{\mathbf{a}}&{}\approx{}&-0.172158753520&{}:{}&0.645541098885&{}:{}&0.526617654634&, \\ P^+_{\mathbf{a}}&{}\approx{}&-0.243795807301&{}:{}&0.641267077682&{}:{}&0.602528729619&, \\ Q_{\mathbf{a}}&{}\approx{}&-0.153100084143&{}:{}&0.671406744368&{}:{}&0.481693339775&, \\ I^\prime_{\mathbf{a}}&{}\approx{}&-0.279854846231&{}:{}&0.635286063680&{}:{}&0.644568782551&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{a}}\)
\(P^-_{\mathbf{a}}\)
\(P^+_{\mathbf{a}}\)
\(Q_{\mathbf{a}}\)
\(I^\prime_{\mathbf{a}}\)
0a (011)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{a}}&{}\approx{}&-0.754600960884&{}:{}&0.814313139868&{}:{}&0.940287821015&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.142482539092&{}:{}&0.857517460908&{}:{}&0.284965078184&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.177250691694&{}:{}&0.307007203691&{}:{}&0.870243488003&, \\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.157976251304&{}:{}&0.612193885840&{}:{}&0.545782365463&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.335323888807&{}:{}&0.445902906223&{}:{}&0.889420982584&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.284846922835&{}:{}&0.847469149469&{}:{}&0.437377773366&, \\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.100724410341&{}:{}&0.606200178336&{}:{}&0.494524232005&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.318670453370&{}:{}&0.343887101817&{}:{}&0.974783351553&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.250265410457&{}:{}&0.938416460254&{}:{}&0.311848950203&, \\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.582262332300&{}:{}&0.417737667700&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.465925826289&{}:{}&0.000000000000&{}:{}&1.465925826289&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.295385137588&{}:{}&1.295385137588&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{a}}}{B^\prime_{\mathbf{a}}}{C^\prime_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{a}}}{B^{\prime\prime}_{\mathbf{a}}}{C^{\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{a}}}{B^{\prime\prime\prime}_{\mathbf{a}}}{C^{\prime\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^*_{\mathbf{a}}}{B^*_{\mathbf{a}}}{C^*_{\mathbf{a}}}\)
0a (011)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{a}}}}&{}\approx{}&1.284457050376&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{a}}}}&{}\approx{}&0.389269535991&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{a}}}}&{}\approx{}&0.484257895385&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-0.754600960884&{}:{}&0.814313139868&{}:{}&0.940287821015&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.142482539092&{}:{}&0.857517460908&{}:{}&0.284965078184&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.177250691694&{}:{}&0.307007203691&{}:{}&0.870243488003&. \end{alignedat} \]
0a (011)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{a}}&{}\approx{}&-0.216094851447&{}:{}&0.642919776480&{}:{}&0.573175074967&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.157976251304&{}:{}&0.612193885840&{}:{}&0.545782365463&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.335323888807&{}:{}&0.445902906223&{}:{}&0.889420982584&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.284846922835&{}:{}&0.847469149469&{}:{}&0.437377773366&. \end{alignedat} \]
0a (011)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{a}}&{}\approx{}&-0.172158753520&{}:{}&0.645541098885&{}:{}&0.526617654634&,\\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.100724410341&{}:{}&0.606200178336&{}:{}&0.494524232005&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.318670453370&{}:{}&0.343887101817&{}:{}&0.974783351553&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.250265410457&{}:{}&0.938416460254&{}:{}&0.311848950203&. \end{alignedat} \]
0a (011)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{a}}&{}\approx{}&-0.243795807301&{}:{}&0.641267077682&{}:{}&0.602528729619&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-0.754600960884&{}:{}&0.814313139868&{}:{}&0.940287821015&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.142482539092&{}:{}&0.857517460908&{}:{}&0.284965078184&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.177250691694&{}:{}&0.307007203691&{}:{}&0.870243488003&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.157976251304&{}:{}&0.612193885840&{}:{}&0.545782365463&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.335323888807&{}:{}&0.445902906223&{}:{}&0.889420982584&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.284846922835&{}:{}&0.847469149469&{}:{}&0.437377773366&, \end{alignedat} \]
0a (011)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{a}}&{}\approx{}&-0.153100084143&{}:{}&0.671406744368&{}:{}&0.481693339775&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.582262332300&{}:{}&0.417737667700&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.465925826289&{}:{}&0.000000000000&{}:{}&1.465925826289&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.295385137588&{}:{}&1.295385137588&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{a}}&{}\approx{}&-0.279854846231&{}:{}&0.635286063680&{}:{}&0.644568782551&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.157976251304&{}:{}&0.612193885840&{}:{}&0.545782365463&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.335323888807&{}:{}&0.445902906223&{}:{}&0.889420982584&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.284846922835&{}:{}&0.847469149469&{}:{}&0.437377773366&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.582262332300&{}:{}&0.417737667700&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.465925826289&{}:{}&0.000000000000&{}:{}&1.465925826289&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.295385137588&{}:{}&1.295385137588&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Hiroyasu Kamo