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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\(\mathbf{1b}\) \((103)\)

Malfatti circles

1b (103)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&, \\ P_{\mathbf{1b}}&{}\approx{}&0.905998181659&{}:{}&-0.247571901500&{}:{}&0.341573719841&, \\ P^-_{\mathbf{1b}}&{}\approx{}&0.924096219046&{}:{}&-0.075041321982&{}:{}&0.150945102935&, \\ P^+_{\mathbf{1b}}&{}\approx{}&0.892167221276&{}:{}&-0.379423966951&{}:{}&0.487256745675&, \\ Q_{\mathbf{1b}}&{}\approx{}&1.126421175455&{}:{}&0.231597427882&{}:{}&-0.358018603337&, \\ I^\prime_{\mathbf{1b}}&{}\approx{}&0.847184126031&{}:{}&-0.582834345953&{}:{}&0.735650219922&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{1b}}\)
\(P^-_{\mathbf{1b}}\)
\(P^+_{\mathbf{1b}}\)
\(Q_{\mathbf{1b}}\)
\(I^\prime_{\mathbf{1b}}\)
1b (103)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1b}}&{}\approx{}&0.965925826289&{}:{}&-0.220258921319&{}:{}&0.254333095030&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.419606348368&{}:{}&-0.258819045103&{}:{}&0.839212696735&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.965925826289&{}:{}&-3.405083415044&{}:{}&2.439157588755&, \\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.691598265489&{}:{}&-0.812235392740&{}:{}&1.120637127251&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.026681170442&{}:{}&-0.413754029503&{}:{}&0.387072859061&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.838864733154&{}:{}&-0.229227101436&{}:{}&0.390362368282&, \\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.615800306227&{}:{}&-0.379834213309&{}:{}&0.764033907082&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.069158888269&{}:{}&-0.243799034087&{}:{}&0.174640145818&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.845952191110&{}:{}&-0.068695628708&{}:{}&0.222743437598&, \\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.831951230074&{}:{}&2.831951230074&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.465925826289&{}:{}&0.000000000000&{}:{}&-0.465925826289&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.829459311299&{}:{}&0.170540688701&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1b}}}{B^\prime_{\mathbf{1b}}}{C^\prime_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^*_{\mathbf{1b}}}{B^*_{\mathbf{1b}}}{C^*_{\mathbf{1b}}}\)
1b (103)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1b}}}}&{}\approx{}&0.161240721226&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1b}}}}&{}\approx{}&0.532039530552&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1b}}}}&{}\approx{}&2.492694063823&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.965925826289&{}:{}&-0.220258921319&{}:{}&0.254333095030&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.419606348368&{}:{}&-0.258819045103&{}:{}&0.839212696735&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.965925826289&{}:{}&-3.405083415044&{}:{}&2.439157588755&. \end{alignedat} \]
1b (103)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1b}}&{}\approx{}&0.905998181659&{}:{}&-0.247571901500&{}:{}&0.341573719841&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.691598265489&{}:{}&-0.812235392740&{}:{}&1.120637127251&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.026681170442&{}:{}&-0.413754029503&{}:{}&0.387072859061&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.838864733154&{}:{}&-0.229227101436&{}:{}&0.390362368282&. \end{alignedat} \]
1b (103)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1b}}&{}\approx{}&0.924096219046&{}:{}&-0.075041321982&{}:{}&0.150945102935&,\\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.615800306227&{}:{}&-0.379834213309&{}:{}&0.764033907082&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.069158888269&{}:{}&-0.243799034087&{}:{}&0.174640145818&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.845952191110&{}:{}&-0.068695628708&{}:{}&0.222743437598&. \end{alignedat} \]
1b (103)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1b}}&{}\approx{}&0.892167221276&{}:{}&-0.379423966951&{}:{}&0.487256745675&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.965925826289&{}:{}&-0.220258921319&{}:{}&0.254333095030&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.419606348368&{}:{}&-0.258819045103&{}:{}&0.839212696735&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.965925826289&{}:{}&-3.405083415044&{}:{}&2.439157588755&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.691598265489&{}:{}&-0.812235392740&{}:{}&1.120637127251&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.026681170442&{}:{}&-0.413754029503&{}:{}&0.387072859061&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.838864733154&{}:{}&-0.229227101436&{}:{}&0.390362368282&, \end{alignedat} \]
1b (103)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1b}}&{}\approx{}&1.126421175455&{}:{}&0.231597427882&{}:{}&-0.358018603337&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.831951230074&{}:{}&2.831951230074&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.465925826289&{}:{}&0.000000000000&{}:{}&-0.465925826289&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.829459311299&{}:{}&0.170540688701&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1b}}&{}\approx{}&0.847184126031&{}:{}&-0.582834345953&{}:{}&0.735650219922&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.691598265489&{}:{}&-0.812235392740&{}:{}&1.120637127251&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.026681170442&{}:{}&-0.413754029503&{}:{}&0.387072859061&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.838864733154&{}:{}&-0.229227101436&{}:{}&0.390362368282&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.831951230074&{}:{}&2.831951230074&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.465925826289&{}:{}&0.000000000000&{}:{}&-0.465925826289&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.829459311299&{}:{}&0.170540688701&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Hiroyasu Kamo