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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2c (130)

Malfatti circles

2c (130)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&, \\ P_{\mathbf{2c}}&{}\approx{}&0.956136725788&{}:{}&0.404339911711&{}:{}&-0.360476637498&, \\ P^-_{\mathbf{2c}}&{}\approx{}&0.909935441570&{}:{}&0.183225268078&{}:{}&-0.093160709648&, \\ P^+_{\mathbf{2c}}&{}\approx{}&0.993838723272&{}:{}&0.584777834148&{}:{}&-0.578616557420&, \\ Q_{\mathbf{2c}}&{}\approx{}&1.143091245162&{}:{}&-0.506408217532&{}:{}&0.363316972370&, \\ I^\prime_{\mathbf{2c}}&{}\approx{}&1.012663821604&{}:{}&0.866680278669&{}:{}&-0.879344100274&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{2c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{2c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{2c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{2c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{2c}}\) Radical center of the Malfatti circles
2c (130)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2c}}&{}\approx{}&1.047494179697&{}:{}&0.307007203691&{}:{}&-0.354501383389&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&5.263802882652&{}:{}&6.263802882652&{}:{}&-10.527605765303&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.470143910508&{}:{}&0.814313139868&{}:{}&-0.284457050376&, \\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.863190734293&{}:{}&1.261133543050&{}:{}&-1.124324277343&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.889420982584&{}:{}&0.445902906223&{}:{}&-0.335323888807&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.184846922835&{}:{}&0.501058987955&{}:{}&-0.685905910790&, \\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.630941014909&{}:{}&0.750805118671&{}:{}&-0.381746133580&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.839838163367&{}:{}&0.246145869911&{}:{}&-0.085984033278&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.158838906697&{}:{}&0.233344652421&{}:{}&-0.392183559117&, \\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&3.539058011260&{}:{}&-2.539058011260&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.758819045103&{}:{}&0.000000000000&{}:{}&0.241180954897&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.795385137588&{}:{}&-0.795385137588&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2c}}}{B^\prime_{\mathbf{2c}}}{C^\prime_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^*_{\mathbf{2c}}}{B^*_{\mathbf{2c}}}{C^*_{\mathbf{2c}}}\)
2c (130)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2c}}}}&{}\approx{}&0.129756511997&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2c}}}}&{}\approx{}&3.853371151129&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2c}}}}&{}\approx{}&0.344169229361&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.047494179697&{}:{}&0.307007203691&{}:{}&-0.354501383389&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&5.263802882652&{}:{}&6.263802882652&{}:{}&-10.527605765303&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.470143910508&{}:{}&0.814313139868&{}:{}&-0.284457050376&. \end{alignedat} \]
2c (130)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2c}}&{}\approx{}&0.956136725788&{}:{}&0.404339911711&{}:{}&-0.360476637498&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.863190734293&{}:{}&1.261133543050&{}:{}&-1.124324277343&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.889420982584&{}:{}&0.445902906223&{}:{}&-0.335323888807&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.184846922835&{}:{}&0.501058987955&{}:{}&-0.685905910790&. \end{alignedat} \]
2c (130)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2c}}&{}\approx{}&0.909935441570&{}:{}&0.183225268078&{}:{}&-0.093160709648&,\\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.630941014909&{}:{}&0.750805118671&{}:{}&-0.381746133580&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.839838163367&{}:{}&0.246145869911&{}:{}&-0.085984033278&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.158838906697&{}:{}&0.233344652421&{}:{}&-0.392183559117&. \end{alignedat} \]
2c (130)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2c}}&{}\approx{}&0.993838723272&{}:{}&0.584777834148&{}:{}&-0.578616557420&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.047494179697&{}:{}&0.307007203691&{}:{}&-0.354501383389&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&5.263802882652&{}:{}&6.263802882652&{}:{}&-10.527605765303&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.470143910508&{}:{}&0.814313139868&{}:{}&-0.284457050376&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.863190734293&{}:{}&1.261133543050&{}:{}&-1.124324277343&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.889420982584&{}:{}&0.445902906223&{}:{}&-0.335323888807&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.184846922835&{}:{}&0.501058987955&{}:{}&-0.685905910790&, \end{alignedat} \]
2c (130)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2c}}&{}\approx{}&1.143091245162&{}:{}&-0.506408217532&{}:{}&0.363316972370&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&3.539058011260&{}:{}&-2.539058011260&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.758819045103&{}:{}&0.000000000000&{}:{}&0.241180954897&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.795385137588&{}:{}&-0.795385137588&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2c}}&{}\approx{}&1.012663821604&{}:{}&0.866680278669&{}:{}&-0.879344100274&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.863190734293&{}:{}&1.261133543050&{}:{}&-1.124324277343&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.889420982584&{}:{}&0.445902906223&{}:{}&-0.335323888807&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.184846922835&{}:{}&0.501058987955&{}:{}&-0.685905910790&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&3.539058011260&{}:{}&-2.539058011260&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.758819045103&{}:{}&0.000000000000&{}:{}&0.241180954897&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.795385137588&{}:{}&-0.795385137588&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)