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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\(\mathbf{3}\) \((022)\)

Malfatti circles

3 (022)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&, \\ P_{\mathbf{3}}&{}\approx{}&0.002154922694&{}:{}&0.803677231266&{}:{}&0.194167846040&, \\ P^-_{\mathbf{3}}&{}\approx{}&-0.069605532935&{}:{}&0.953823538808&{}:{}&0.115781994127&, \\ P^+_{\mathbf{3}}&{}\approx{}&0.044713810241&{}:{}&0.714630140326&{}:{}&0.240656049433&, \\ Q_{\mathbf{3}}&{}\approx{}&-0.021888776929&{}:{}&0.620497363217&{}:{}&0.401391413712&, \\ I^\prime_{\mathbf{3}}&{}\approx{}&0.025100357596&{}:{}&0.637947892917&{}:{}&0.336951749487&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{3}}\)
\(P^-_{\mathbf{3}}\)
\(P^+_{\mathbf{3}}\)
\(Q_{\mathbf{3}}\)
\(I^\prime_{\mathbf{3}}\)
3 (022)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3}}&{}\approx{}&-0.346352670420&{}:{}&0.624844448887&{}:{}&0.721508221533&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.061895620044&{}:{}&0.814313139868&{}:{}&0.123791240088&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.230997598770&{}:{}&0.400099577495&{}:{}&0.368902823735&, \\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.097631072938&{}:{}&0.726779514551&{}:{}&0.175589412511&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.005621780360&{}:{}&0.487831517751&{}:{}&0.506546701889&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.002109042109&{}:{}&0.786566092485&{}:{}&0.211324865405&, \\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.063479042966&{}:{}&0.835145019262&{}:{}&0.101375937772&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.405273068521&{}:{}&0.731140969236&{}:{}&0.674132099284&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.067629565656&{}:{}&0.926746322053&{}:{}&0.140883243603&, \\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.607206358682&{}:{}&0.392793641318&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.057677535825&{}:{}&0.000000000000&{}:{}&1.057677535825&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.036566092485&{}:{}&1.036566092485&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3}}}{B^\prime_{\mathbf{3}}}{C^\prime_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3}}}{B^{\prime\prime}_{\mathbf{3}}}{C^{\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3}}}{B^{\prime\prime\prime}_{\mathbf{3}}}{C^{\prime\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^*_{\mathbf{3}}}{B^*_{\mathbf{3}}}{C^*_{\mathbf{3}}}\)
3 (022)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3}}}}&{}\approx{}&1.707106781187&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3}}}}&{}\approx{}&0.292893218813&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3}}}}&{}\approx{}&1.093092373804&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.346352670420&{}:{}&0.624844448887&{}:{}&0.721508221533&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.061895620044&{}:{}&0.814313139868&{}:{}&0.123791240088&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.230997598770&{}:{}&0.400099577495&{}:{}&0.368902823735&. \end{alignedat} \]
3 (022)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3}}&{}\approx{}&0.002154922694&{}:{}&0.803677231266&{}:{}&0.194167846040&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.097631072938&{}:{}&0.726779514551&{}:{}&0.175589412511&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.005621780360&{}:{}&0.487831517751&{}:{}&0.506546701889&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.002109042109&{}:{}&0.786566092485&{}:{}&0.211324865405&. \end{alignedat} \]
3 (022)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3}}&{}\approx{}&-0.069605532935&{}:{}&0.953823538808&{}:{}&0.115781994127&,\\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.063479042966&{}:{}&0.835145019262&{}:{}&0.101375937772&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.405273068521&{}:{}&0.731140969236&{}:{}&0.674132099284&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.067629565656&{}:{}&0.926746322053&{}:{}&0.140883243603&. \end{alignedat} \]
3 (022)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3}}&{}\approx{}&0.044713810241&{}:{}&0.714630140326&{}:{}&0.240656049433&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.346352670420&{}:{}&0.624844448887&{}:{}&0.721508221533&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.061895620044&{}:{}&0.814313139868&{}:{}&0.123791240088&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.230997598770&{}:{}&0.400099577495&{}:{}&0.368902823735&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.097631072938&{}:{}&0.726779514551&{}:{}&0.175589412511&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.005621780360&{}:{}&0.487831517751&{}:{}&0.506546701889&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.002109042109&{}:{}&0.786566092485&{}:{}&0.211324865405&, \end{alignedat} \]
3 (022)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3}}&{}\approx{}&-0.021888776929&{}:{}&0.620497363217&{}:{}&0.401391413712&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.607206358682&{}:{}&0.392793641318&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.057677535825&{}:{}&0.000000000000&{}:{}&1.057677535825&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.036566092485&{}:{}&1.036566092485&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3}}&{}\approx{}&0.025100357596&{}:{}&0.637947892917&{}:{}&0.336951749487&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.097631072938&{}:{}&0.726779514551&{}:{}&0.175589412511&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.005621780360&{}:{}&0.487831517751&{}:{}&0.506546701889&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.002109042109&{}:{}&0.786566092485&{}:{}&0.211324865405&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.607206358682&{}:{}&0.392793641318&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.057677535825&{}:{}&0.000000000000&{}:{}&1.057677535825&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.036566092485&{}:{}&1.036566092485&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)

Hiroyasu Kamo