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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[Lob & Richmond]

\(\mathbf{4}\) \((200)\)

Malfatti circles

4 (200)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&, \\ P_{\mathbf{4}}&{}\approx{}&0.998627535788&{}:{}&0.000576739540&{}:{}&0.000795724672&, \\ P^-_{\mathbf{4}}&{}\approx{}&1.019041582560&{}:{}&-0.008899014074&{}:{}&-0.010142568485&, \\ P^+_{\mathbf{4}}&{}\approx{}&0.979219932258&{}:{}&0.009585324234&{}:{}&0.011194743507&, \\ Q_{\mathbf{4}}&{}\approx{}&1.098669811316&{}:{}&-0.038756874475&{}:{}&-0.059912936841&, \\ I^\prime_{\mathbf{4}}&{}\approx{}&0.933229191125&{}:{}&0.029515947116&{}:{}&0.037254861759&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{4}}\)
\(P^-_{\mathbf{4}}\)
\(P^+_{\mathbf{4}}\)
\(Q_{\mathbf{4}}\)
\(I^\prime_{\mathbf{4}}\)
4 (200)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4}}&{}\approx{}&0.980327266636&{}:{}&0.009130147329&{}:{}&0.010542586036&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.726779514551&{}:{}&-1.180338543653&{}:{}&1.453559029102&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.135027805015&{}:{}&1.965925826289&{}:{}&-2.100953631304&, \\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.886144284064&{}:{}&0.047844667040&{}:{}&0.066011048897&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.981042405207&{}:{}&0.018175882275&{}:{}&0.000781712519&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.978501540222&{}:{}&0.000565116130&{}:{}&0.020933343648&, \\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.404027016596&{}:{}&0.656166899063&{}:{}&0.747860117533&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.000640089468&{}:{}&0.009319328097&{}:{}&-0.009959417565&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.997982657437&{}:{}&-0.008715112186&{}:{}&0.010732454749&, \\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.392793641318&{}:{}&0.607206358682&,\\B^*_{\mathbf{4}}&{}\approx{}&1.057677535825&{}:{}&0.000000000000&{}:{}&-0.057677535825&,\\C^*_{\mathbf{4}}&{}\approx{}&1.036566092485&{}:{}&-0.036566092485&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4}}}{B^\prime_{\mathbf{4}}}{C^\prime_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4}}}{B^{\prime\prime}_{\mathbf{4}}}{C^{\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4}}}{B^{\prime\prime\prime}_{\mathbf{4}}}{C^{\prime\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^*_{\mathbf{4}}}{B^*_{\mathbf{4}}}{C^*_{\mathbf{4}}}\)
4 (200)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4}}}}&{}\approx{}&0.024944026382&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4}}}}&{}\approx{}&3.439157588755&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4}}}}&{}\approx{}&5.371009241334&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.980327266636&{}:{}&0.009130147329&{}:{}&0.010542586036&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.726779514551&{}:{}&-1.180338543653&{}:{}&1.453559029102&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.135027805015&{}:{}&1.965925826289&{}:{}&-2.100953631304&. \end{alignedat} \]
4 (200)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4}}&{}\approx{}&0.998627535788&{}:{}&0.000576739540&{}:{}&0.000795724672&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.886144284064&{}:{}&0.047844667040&{}:{}&0.066011048897&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.981042405207&{}:{}&0.018175882275&{}:{}&0.000781712519&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.978501540222&{}:{}&0.000565116130&{}:{}&0.020933343648&. \end{alignedat} \]
4 (200)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4}}&{}\approx{}&1.019041582560&{}:{}&-0.008899014074&{}:{}&-0.010142568485&,\\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.404027016596&{}:{}&0.656166899063&{}:{}&0.747860117533&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.000640089468&{}:{}&0.009319328097&{}:{}&-0.009959417565&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.997982657437&{}:{}&-0.008715112186&{}:{}&0.010732454749&. \end{alignedat} \]
4 (200)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4}}&{}\approx{}&0.979219932258&{}:{}&0.009585324234&{}:{}&0.011194743507&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.980327266636&{}:{}&0.009130147329&{}:{}&0.010542586036&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.726779514551&{}:{}&-1.180338543653&{}:{}&1.453559029102&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.135027805015&{}:{}&1.965925826289&{}:{}&-2.100953631304&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.886144284064&{}:{}&0.047844667040&{}:{}&0.066011048897&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.981042405207&{}:{}&0.018175882275&{}:{}&0.000781712519&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.978501540222&{}:{}&0.000565116130&{}:{}&0.020933343648&, \end{alignedat} \]
4 (200)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4}}&{}\approx{}&1.098669811316&{}:{}&-0.038756874475&{}:{}&-0.059912936841&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.392793641318&{}:{}&0.607206358682&,\\B^*_{\mathbf{4}}&{}\approx{}&1.057677535825&{}:{}&0.000000000000&{}:{}&-0.057677535825&,\\C^*_{\mathbf{4}}&{}\approx{}&1.036566092485&{}:{}&-0.036566092485&{}:{}&0.000000000000&. \end{alignedat} \]
4 (200)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4}}&{}\approx{}&0.933229191125&{}:{}&0.029515947116&{}:{}&0.037254861759&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.886144284064&{}:{}&0.047844667040&{}:{}&0.066011048897&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.981042405207&{}:{}&0.018175882275&{}:{}&0.000781712519&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.978501540222&{}:{}&0.000565116130&{}:{}&0.020933343648&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.392793641318&{}:{}&0.607206358682&,\\B^*_{\mathbf{4}}&{}\approx{}&1.057677535825&{}:{}&0.000000000000&{}:{}&-0.057677535825&,\\C^*_{\mathbf{4}}&{}\approx{}&1.036566092485&{}:{}&-0.036566092485&{}:{}&0.000000000000&. \end{alignedat} \]
4 (200)

Hiroyasu Kamo