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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\(\mathbf{5c}\) \((312)\)

Malfatti circles

5c (312)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&, \\ P_{\mathbf{5c}}&{}\approx{}&-0.087838897224&{}:{}&-0.037146122506&{}:{}&1.124985019730&, \\ P^-_{\mathbf{5c}}&{}\approx{}&-0.013038129853&{}:{}&0.086496141884&{}:{}&0.926541987969&, \\ P^+_{\mathbf{5c}}&{}\approx{}&-0.171219461465&{}:{}&-0.174970400666&{}:{}&1.346189862131&, \\ Q_{\mathbf{5c}}&{}\approx{}&-0.771898173680&{}:{}&0.341963583313&{}:{}&1.429934590367&, \\ I^\prime_{\mathbf{5c}}&{}\approx{}&-0.311987357086&{}:{}&-0.267011898531&{}:{}&1.578999255617&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{5c}}\)
\(P^-_{\mathbf{5c}}\)
\(P^+_{\mathbf{5c}}\)
\(Q_{\mathbf{5c}}\)
\(I^\prime_{\mathbf{5c}}\)
5c (312)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5c}}&{}\approx{}&-2.250382876665&{}:{}&-21.010805202868&{}:{}&24.261188079533&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.448287736084&{}:{}&0.551712263916&{}:{}&0.896575472168&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.095682338286&{}:{}&-0.165726671298&{}:{}&1.261409009584&, \\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.157704332773&{}:{}&-0.039531797475&{}:{}&1.197236130248&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.112544936467&{}:{}&-0.328859402310&{}:{}&1.441404338777&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.512511024384&{}:{}&-0.216735386019&{}:{}&1.729246410402&, \\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.074548920027&{}:{}&0.091748112049&{}:{}&0.982800807978&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.016467010125&{}:{}&-0.153745011840&{}:{}&1.170212021965&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.060919864928&{}:{}&0.404147936857&{}:{}&0.656771928072&, \\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.192992796309&{}:{}&0.807007203691&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.173032607476&{}:{}&0.000000000000&{}:{}&2.173032607476&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.795385137588&{}:{}&-0.795385137588&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5c}}}{B^\prime_{\mathbf{5c}}}{C^\prime_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^*_{\mathbf{5c}}}{B^*_{\mathbf{5c}}}{C^*_{\mathbf{5c}}}\)
5c (312)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5c}}}}&{}\approx{}&-8.880211163101&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.328169399224&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.070044333012&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-2.250382876665&{}:{}&-21.010805202868&{}:{}&24.261188079533&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.448287736084&{}:{}&0.551712263916&{}:{}&0.896575472168&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.095682338286&{}:{}&-0.165726671298&{}:{}&1.261409009584&. \end{alignedat} \]
5c (312)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5c}}&{}\approx{}&-0.087838897224&{}:{}&-0.037146122506&{}:{}&1.124985019730&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.157704332773&{}:{}&-0.039531797475&{}:{}&1.197236130248&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.112544936467&{}:{}&-0.328859402310&{}:{}&1.441404338777&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.512511024384&{}:{}&-0.216735386019&{}:{}&1.729246410402&. \end{alignedat} \]
5c (312)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5c}}&{}\approx{}&-0.013038129853&{}:{}&0.086496141884&{}:{}&0.926541987969&,\\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.074548920027&{}:{}&0.091748112049&{}:{}&0.982800807978&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.016467010125&{}:{}&-0.153745011840&{}:{}&1.170212021965&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.060919864928&{}:{}&0.404147936857&{}:{}&0.656771928072&. \end{alignedat} \]
5c (312)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5c}}&{}\approx{}&-0.171219461465&{}:{}&-0.174970400666&{}:{}&1.346189862131&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-2.250382876665&{}:{}&-21.010805202868&{}:{}&24.261188079533&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.448287736084&{}:{}&0.551712263916&{}:{}&0.896575472168&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.095682338286&{}:{}&-0.165726671298&{}:{}&1.261409009584&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.157704332773&{}:{}&-0.039531797475&{}:{}&1.197236130248&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.112544936467&{}:{}&-0.328859402310&{}:{}&1.441404338777&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.512511024384&{}:{}&-0.216735386019&{}:{}&1.729246410402&, \end{alignedat} \]
5c (312)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5c}}&{}\approx{}&-0.771898173680&{}:{}&0.341963583313&{}:{}&1.429934590367&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.192992796309&{}:{}&0.807007203691&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.173032607476&{}:{}&0.000000000000&{}:{}&2.173032607476&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.795385137588&{}:{}&-0.795385137588&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5c}}&{}\approx{}&-0.311987357086&{}:{}&-0.267011898531&{}:{}&1.578999255617&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.157704332773&{}:{}&-0.039531797475&{}:{}&1.197236130248&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.112544936467&{}:{}&-0.328859402310&{}:{}&1.441404338777&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.512511024384&{}:{}&-0.216735386019&{}:{}&1.729246410402&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.192992796309&{}:{}&0.807007203691&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.173032607476&{}:{}&0.000000000000&{}:{}&2.173032607476&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.795385137588&{}:{}&-0.795385137588&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)

Hiroyasu Kamo