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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\(\mathbf{7c}\) \((332)\)

Malfatti circles

7c (332)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&, \\ P_{\mathbf{7c}}&{}\approx{}&-0.028798816464&{}:{}&-0.035130097818&{}:{}&1.063928914282&, \\ P^-_{\mathbf{7c}}&{}\approx{}&0.012093838750&{}:{}&0.035265599803&{}:{}&0.952640561447&, \\ P^+_{\mathbf{7c}}&{}\approx{}&-0.072238554233&{}:{}&-0.109910535252&{}:{}&1.182149089485&, \\ Q_{\mathbf{7c}}&{}\approx{}&0.204136545810&{}:{}&0.153595913504&{}:{}&0.642267540686&, \\ I^\prime_{\mathbf{7c}}&{}\approx{}&-0.162808244672&{}:{}&-0.236651394056&{}:{}&1.399459638728&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{7c}}\)
\(P^-_{\mathbf{7c}}\)
\(P^+_{\mathbf{7c}}\)
\(Q_{\mathbf{7c}}\)
\(I^\prime_{\mathbf{7c}}\)
7c (332)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7c}}&{}\approx{}&0.529856089492&{}:{}&-3.039058011260&{}:{}&3.509201921768&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.531752075083&{}:{}&0.468247924917&{}:{}&1.063504150166&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.047494179697&{}:{}&-0.082262332300&{}:{}&1.129756511997&, \\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.087199972033&{}:{}&-0.037124305311&{}:{}&1.124324277343&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.032278125442&{}:{}&-0.160188620509&{}:{}&1.192466745950&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.284846922835&{}:{}&-0.347469149469&{}:{}&1.632316072304&, \\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.042251430014&{}:{}&0.037205580111&{}:{}&1.005045849903&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.013507105779&{}:{}&-0.077471749102&{}:{}&1.063964643324&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.094886282107&{}:{}&0.276688131919&{}:{}&0.628425585974&, \\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.192992796309&{}:{}&0.807007203691&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.241180954897&{}:{}&0.000000000000&{}:{}&0.758819045103&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.570640266196&{}:{}&0.429359733804&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7c}}}{B^\prime_{\mathbf{7c}}}{C^\prime_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^*_{\mathbf{7c}}}{B^*_{\mathbf{7c}}}{C^*_{\mathbf{7c}}}\)
7c (332)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7c}}}}&{}\approx{}&-1.284457050376&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.389269535991&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.034768152602&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.529856089492&{}:{}&-3.039058011260&{}:{}&3.509201921768&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.531752075083&{}:{}&0.468247924917&{}:{}&1.063504150166&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.047494179697&{}:{}&-0.082262332300&{}:{}&1.129756511997&. \end{alignedat} \]
7c (332)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7c}}&{}\approx{}&-0.028798816464&{}:{}&-0.035130097818&{}:{}&1.063928914282&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.087199972033&{}:{}&-0.037124305311&{}:{}&1.124324277343&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.032278125442&{}:{}&-0.160188620509&{}:{}&1.192466745950&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.284846922835&{}:{}&-0.347469149469&{}:{}&1.632316072304&. \end{alignedat} \]
7c (332)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7c}}&{}\approx{}&0.012093838750&{}:{}&0.035265599803&{}:{}&0.952640561447&,\\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.042251430014&{}:{}&0.037205580111&{}:{}&1.005045849903&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.013507105779&{}:{}&-0.077471749102&{}:{}&1.063964643324&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.094886282107&{}:{}&0.276688131919&{}:{}&0.628425585974&. \end{alignedat} \]
7c (332)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7c}}&{}\approx{}&-0.072238554233&{}:{}&-0.109910535252&{}:{}&1.182149089485&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.529856089492&{}:{}&-3.039058011260&{}:{}&3.509201921768&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.531752075083&{}:{}&0.468247924917&{}:{}&1.063504150166&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.047494179697&{}:{}&-0.082262332300&{}:{}&1.129756511997&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.087199972033&{}:{}&-0.037124305311&{}:{}&1.124324277343&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.032278125442&{}:{}&-0.160188620509&{}:{}&1.192466745950&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.284846922835&{}:{}&-0.347469149469&{}:{}&1.632316072304&, \end{alignedat} \]
7c (332)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7c}}&{}\approx{}&0.204136545810&{}:{}&0.153595913504&{}:{}&0.642267540686&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.192992796309&{}:{}&0.807007203691&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.241180954897&{}:{}&0.000000000000&{}:{}&0.758819045103&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.570640266196&{}:{}&0.429359733804&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7c}}&{}\approx{}&-0.162808244672&{}:{}&-0.236651394056&{}:{}&1.399459638728&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.087199972033&{}:{}&-0.037124305311&{}:{}&1.124324277343&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.032278125442&{}:{}&-0.160188620509&{}:{}&1.192466745950&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.284846922835&{}:{}&-0.347469149469&{}:{}&1.632316072304&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.192992796309&{}:{}&0.807007203691&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.241180954897&{}:{}&0.000000000000&{}:{}&0.758819045103&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.570640266196&{}:{}&0.429359733804&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Hiroyasu Kamo