Derousseau's Generalization of the Malfatti circles

Half of an Equilateral Triangle

\(A=30\degree\), \(B=60\degree\), \(C=90\degree\).


[Top] > Half of an Equilateral Triangle > 6c (330)
[Other solutions]
[Guy]
[Lob & Richmond]

\(\mathbf{6c}\) \((330)\)

Malfatti circles

6c (330)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&, \\ P_{\mathbf{6c}}&{}\approx{}&0.225785148785&{}:{}&0.859338943864&{}:{}&-0.085124092649&, \\ P^-_{\mathbf{6c}}&{}\approx{}&0.137632042349&{}:{}&0.742855517425&{}:{}&0.119512440226&, \\ P^+_{\mathbf{6c}}&{}\approx{}&0.302133172204&{}:{}&0.960223413275&{}:{}&-0.262356585479&, \\ Q_{\mathbf{6c}}&{}\approx{}&0.377802362094&{}:{}&0.502117975910&{}:{}&0.120079661996&, \\ I^\prime_{\mathbf{6c}}&{}\approx{}&0.370483073989&{}:{}&0.951224977995&{}:{}&-0.321708051985&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{6c}}\)
\(P^-_{\mathbf{6c}}\)
\(P^+_{\mathbf{6c}}\)
\(Q_{\mathbf{6c}}\)
\(I^\prime_{\mathbf{6c}}\)
6c (330)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6c}}&{}\approx{}&1.258819045103&{}:{}&1.673032607476&{}:{}&-1.931851652578&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.165726671298&{}:{}&1.165726671298&{}:{}&-0.331453342597&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.258819045103&{}:{}&0.448287736084&{}:{}&0.292893218813&, \\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.202066430806&{}:{}&0.885665509450&{}:{}&-0.087731940256&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.470143910508&{}:{}&0.707106781187&{}:{}&-0.177250691694&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.325806343210&{}:{}&1.240019905583&{}:{}&-0.565826248794&, \\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.109102601687&{}:{}&0.767431166621&{}:{}&0.123466231692&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.312754422633&{}:{}&0.415666055605&{}:{}&0.271579521762&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.205645020102&{}:{}&1.109948927634&{}:{}&-0.315593947736&, \\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.807007203691&{}:{}&0.192992796309&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.758819045103&{}:{}&0.000000000000&{}:{}&0.241180954897&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.429359733804&{}:{}&0.570640266196&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6c}}}{B^\prime_{\mathbf{6c}}}{C^\prime_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^*_{\mathbf{6c}}}{B^*_{\mathbf{6c}}}{C^*_{\mathbf{6c}}}\)
6c (330)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6c}}}}&{}\approx{}&0.707106781187&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6c}}}}&{}\approx{}&0.121320343560&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6c}}}}&{}\approx{}&0.189468690982&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.258819045103&{}:{}&1.673032607476&{}:{}&-1.931851652578&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.165726671298&{}:{}&1.165726671298&{}:{}&-0.331453342597&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.258819045103&{}:{}&0.448287736084&{}:{}&0.292893218813&. \end{alignedat} \]
6c (330)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6c}}&{}\approx{}&0.225785148785&{}:{}&0.859338943864&{}:{}&-0.085124092649&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.202066430806&{}:{}&0.885665509450&{}:{}&-0.087731940256&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.470143910508&{}:{}&0.707106781187&{}:{}&-0.177250691694&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.325806343210&{}:{}&1.240019905583&{}:{}&-0.565826248794&. \end{alignedat} \]
6c (330)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6c}}&{}\approx{}&0.137632042349&{}:{}&0.742855517425&{}:{}&0.119512440226&,\\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.109102601687&{}:{}&0.767431166621&{}:{}&0.123466231692&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.312754422633&{}:{}&0.415666055605&{}:{}&0.271579521762&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.205645020102&{}:{}&1.109948927634&{}:{}&-0.315593947736&. \end{alignedat} \]
6c (330)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6c}}&{}\approx{}&0.302133172204&{}:{}&0.960223413275&{}:{}&-0.262356585479&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.258819045103&{}:{}&1.673032607476&{}:{}&-1.931851652578&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.165726671298&{}:{}&1.165726671298&{}:{}&-0.331453342597&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.258819045103&{}:{}&0.448287736084&{}:{}&0.292893218813&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.202066430806&{}:{}&0.885665509450&{}:{}&-0.087731940256&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.470143910508&{}:{}&0.707106781187&{}:{}&-0.177250691694&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.325806343210&{}:{}&1.240019905583&{}:{}&-0.565826248794&, \end{alignedat} \]
6c (330)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6c}}&{}\approx{}&0.377802362094&{}:{}&0.502117975910&{}:{}&0.120079661996&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.807007203691&{}:{}&0.192992796309&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.758819045103&{}:{}&0.000000000000&{}:{}&0.241180954897&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.429359733804&{}:{}&0.570640266196&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6c}}&{}\approx{}&0.370483073989&{}:{}&0.951224977995&{}:{}&-0.321708051985&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.202066430806&{}:{}&0.885665509450&{}:{}&-0.087731940256&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.470143910508&{}:{}&0.707106781187&{}:{}&-0.177250691694&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.325806343210&{}:{}&1.240019905583&{}:{}&-0.565826248794&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.807007203691&{}:{}&0.192992796309&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.758819045103&{}:{}&0.000000000000&{}:{}&0.241180954897&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.429359733804&{}:{}&0.570640266196&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Hiroyasu Kamo