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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\(\mathbf{1c}\) \((112)\)

Malfatti circles

1c (112)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&, \\ P_{\mathbf{1c}}&{}\approx{}&-0.124978685815&{}:{}&-0.475669247743&{}:{}&1.600647933559&, \\ P^-_{\mathbf{1c}}&{}\approx{}&0.155324995258&{}:{}&0.058559649996&{}:{}&0.786115354746&, \\ P^+_{\mathbf{1c}}&{}\approx{}&-0.574178390070&{}:{}&-1.331795938442&{}:{}&2.905974328512&, \\ Q_{\mathbf{1c}}&{}\approx{}&2.098379994211&{}:{}&2.788850523710&{}:{}&-3.887230517921&, \\ I^\prime_{\mathbf{1c}}&{}\approx{}&-0.669534957192&{}:{}&-1.719048506222&{}:{}&3.388583463414&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{1c}}\)
\(P^-_{\mathbf{1c}}\)
\(P^+_{\mathbf{1c}}\)
\(Q_{\mathbf{1c}}\)
\(I^\prime_{\mathbf{1c}}\)
1c (112)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1c}}&{}\approx{}&0.692992796309&{}:{}&-1.984525761239&{}:{}&2.291532964930&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-0.814313139868&{}:{}&0.185686860132&{}:{}&1.628626279737&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-3.039058011260&{}:{}&-5.263802882652&{}:{}&9.302860893912&, \\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.284457050376&{}:{}&-0.965925826289&{}:{}&3.250382876665&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.328808412459&{}:{}&-2.882361696309&{}:{}&4.211170108768&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.188104379956&{}:{}&-0.715925826289&{}:{}&1.904030206245&, \\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.436847386571&{}:{}&0.099613791793&{}:{}&1.337233594778&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.312754422633&{}:{}&-0.895635873797&{}:{}&1.582881451164&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.213504891485&{}:{}&0.080494267500&{}:{}&0.706000841015&, \\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-2.539058011260&{}:{}&3.539058011260&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.173032607476&{}:{}&0.000000000000&{}:{}&2.173032607476&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.429359733804&{}:{}&0.570640266196&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1c}}}{B^\prime_{\mathbf{1c}}}{C^\prime_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^*_{\mathbf{1c}}}{B^*_{\mathbf{1c}}}{C^*_{\mathbf{1c}}}\)
1c (112)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.838759278774&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.596118591655&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1c}}}}&{}\approx{}&-2.224744871392&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.692992796309&{}:{}&-1.984525761239&{}:{}&2.291532964930&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-0.814313139868&{}:{}&0.185686860132&{}:{}&1.628626279737&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-3.039058011260&{}:{}&-5.263802882652&{}:{}&9.302860893912&. \end{alignedat} \]
1c (112)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1c}}&{}\approx{}&-0.124978685815&{}:{}&-0.475669247743&{}:{}&1.600647933559&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.284457050376&{}:{}&-0.965925826289&{}:{}&3.250382876665&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.328808412459&{}:{}&-2.882361696309&{}:{}&4.211170108768&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.188104379956&{}:{}&-0.715925826289&{}:{}&1.904030206245&. \end{alignedat} \]
1c (112)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1c}}&{}\approx{}&0.155324995258&{}:{}&0.058559649996&{}:{}&0.786115354746&,\\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.436847386571&{}:{}&0.099613791793&{}:{}&1.337233594778&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.312754422633&{}:{}&-0.895635873797&{}:{}&1.582881451164&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.213504891485&{}:{}&0.080494267500&{}:{}&0.706000841015&. \end{alignedat} \]
1c (112)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1c}}&{}\approx{}&-0.574178390070&{}:{}&-1.331795938442&{}:{}&2.905974328512&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.692992796309&{}:{}&-1.984525761239&{}:{}&2.291532964930&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-0.814313139868&{}:{}&0.185686860132&{}:{}&1.628626279737&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-3.039058011260&{}:{}&-5.263802882652&{}:{}&9.302860893912&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.284457050376&{}:{}&-0.965925826289&{}:{}&3.250382876665&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.328808412459&{}:{}&-2.882361696309&{}:{}&4.211170108768&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.188104379956&{}:{}&-0.715925826289&{}:{}&1.904030206245&, \end{alignedat} \]
1c (112)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1c}}&{}\approx{}&2.098379994211&{}:{}&2.788850523710&{}:{}&-3.887230517921&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-2.539058011260&{}:{}&3.539058011260&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.173032607476&{}:{}&0.000000000000&{}:{}&2.173032607476&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.429359733804&{}:{}&0.570640266196&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1c}}&{}\approx{}&-0.669534957192&{}:{}&-1.719048506222&{}:{}&3.388583463414&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.284457050376&{}:{}&-0.965925826289&{}:{}&3.250382876665&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.328808412459&{}:{}&-2.882361696309&{}:{}&4.211170108768&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.188104379956&{}:{}&-0.715925826289&{}:{}&1.904030206245&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-2.539058011260&{}:{}&3.539058011260&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.173032607476&{}:{}&0.000000000000&{}:{}&2.173032607476&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.429359733804&{}:{}&0.570640266196&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Hiroyasu Kamo