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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\(\mathbf{0c}\) \((110)\)

Malfatti circles

0c (110)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&, \\ P_{\mathbf{c}}&{}\approx{}&0.883134575237&{}:{}&1.077287466089&{}:{}&-0.960422041326&, \\ P^-_{\mathbf{c}}&{}\approx{}&0.791900803210&{}:{}&0.833802984173&{}:{}&-0.625703787382&, \\ P^+_{\mathbf{c}}&{}\approx{}&0.949348443703&{}:{}&1.253998886660&{}:{}&-1.203347330362&, \\ Q_{\mathbf{c}}&{}\approx{}&0.824671604353&{}:{}&0.620497363217&{}:{}&-0.445168967570&, \\ I^\prime_{\mathbf{c}}&{}\approx{}&1.021700112433&{}:{}&1.485101423462&{}:{}&-1.506801535895&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{c}}\)
\(P^-_{\mathbf{c}}\)
\(P^+_{\mathbf{c}}\)
\(Q_{\mathbf{c}}\)
\(I^\prime_{\mathbf{c}}\)
0c (110)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{c}}&{}\approx{}&1.095682338286&{}:{}&0.618500357455&{}:{}&-0.714182695741&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.448287736084&{}:{}&1.448287736084&{}:{}&-0.896575472168&,\\C^\prime_{\mathbf{c}}&{}\approx{}&3.250382876665&{}:{}&5.629828286436&{}:{}&-7.880211163101&, \\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.787908377764&{}:{}&1.955100465014&{}:{}&-1.743008842778&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.308968663182&{}:{}&1.114554309167&{}:{}&-1.423522972348&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.808638614765&{}:{}&0.986413926833&{}:{}&-0.795052541598&, \\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.553613173284&{}:{}&1.788563694392&{}:{}&-1.342176867676&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.291389424022&{}:{}&0.728974806348&{}:{}&-1.020364230369&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.713725972751&{}:{}&0.751491655961&{}:{}&-0.465217628712&, \\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&3.539058011260&{}:{}&-2.539058011260&,\\B^*_{\mathbf{c}}&{}\approx{}&2.173032607476&{}:{}&0.000000000000&{}:{}&-1.173032607476&,\\C^*_{\mathbf{c}}&{}\approx{}&0.570640266196&{}:{}&0.429359733804&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{c}}}{B^\prime_{\mathbf{c}}}{C^\prime_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{c}}}{B^{\prime\prime}_{\mathbf{c}}}{C^{\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{c}}}{B^{\prime\prime\prime}_{\mathbf{c}}}{C^{\prime\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^*_{\mathbf{c}}}{B^*_{\mathbf{c}}}{C^*_{\mathbf{c}}}\)
0c (110)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{c}}}}&{}\approx{}&0.261409009584&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{c}}}}&{}\approx{}&0.328169399224&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{c}}}}&{}\approx{}&2.379445409771&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.366025403784&{}:{}&2.366025403784&{}:{}&-2.732050807569&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.095682338286&{}:{}&0.618500357455&{}:{}&-0.714182695741&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.448287736084&{}:{}&1.448287736084&{}:{}&-0.896575472168&,\\C^\prime_{\mathbf{c}}&{}\approx{}&3.250382876665&{}:{}&5.629828286436&{}:{}&-7.880211163101&. \end{alignedat} \]
0c (110)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{c}}&{}\approx{}&0.883134575237&{}:{}&1.077287466089&{}:{}&-0.960422041326&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.787908377764&{}:{}&1.955100465014&{}:{}&-1.743008842778&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.308968663182&{}:{}&1.114554309167&{}:{}&-1.423522972348&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.808638614765&{}:{}&0.986413926833&{}:{}&-0.795052541598&. \end{alignedat} \]
0c (110)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{c}}&{}\approx{}&0.791900803210&{}:{}&0.833802984173&{}:{}&-0.625703787382&,\\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.553613173284&{}:{}&1.788563694392&{}:{}&-1.342176867676&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.291389424022&{}:{}&0.728974806348&{}:{}&-1.020364230369&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.713725972751&{}:{}&0.751491655961&{}:{}&-0.465217628712&. \end{alignedat} \]
0c (110)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{c}}&{}\approx{}&0.949348443703&{}:{}&1.253998886660&{}:{}&-1.203347330362&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.095682338286&{}:{}&0.618500357455&{}:{}&-0.714182695741&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.448287736084&{}:{}&1.448287736084&{}:{}&-0.896575472168&,\\C^\prime_{\mathbf{c}}&{}\approx{}&3.250382876665&{}:{}&5.629828286436&{}:{}&-7.880211163101&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.787908377764&{}:{}&1.955100465014&{}:{}&-1.743008842778&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.308968663182&{}:{}&1.114554309167&{}:{}&-1.423522972348&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.808638614765&{}:{}&0.986413926833&{}:{}&-0.795052541598&, \end{alignedat} \]
0c (110)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{c}}&{}\approx{}&0.824671604353&{}:{}&0.620497363217&{}:{}&-0.445168967570&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&3.539058011260&{}:{}&-2.539058011260&,\\B^*_{\mathbf{c}}&{}\approx{}&2.173032607476&{}:{}&0.000000000000&{}:{}&-1.173032607476&,\\C^*_{\mathbf{c}}&{}\approx{}&0.570640266196&{}:{}&0.429359733804&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{c}}&{}\approx{}&1.021700112433&{}:{}&1.485101423462&{}:{}&-1.506801535895&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.787908377764&{}:{}&1.955100465014&{}:{}&-1.743008842778&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.308968663182&{}:{}&1.114554309167&{}:{}&-1.423522972348&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.808638614765&{}:{}&0.986413926833&{}:{}&-0.795052541598&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&3.539058011260&{}:{}&-2.539058011260&,\\B^*_{\mathbf{c}}&{}\approx{}&2.173032607476&{}:{}&0.000000000000&{}:{}&-1.173032607476&,\\C^*_{\mathbf{c}}&{}\approx{}&0.570640266196&{}:{}&0.429359733804&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Hiroyasu Kamo