Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 108° Top Angle

\(A=36\degree\), \(B=36\degree\), \(C=108\degree\).

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[Top] > Isosceles Triangle with 108° Top Angle > 1b (103)

1b(103)

Malfatti circles

1b (103)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{1b}}&{}\approx{}&0.900381905509&{}:{}&-0.112933240919&{}:{}&0.212551335411&, \\ P^-_{\mathbf{1b}}&{}\approx{}&0.947343589942&{}:{}&-0.028922059966&{}:{}&0.081578470024&, \\ P^+_{\mathbf{1b}}&{}\approx{}&0.865142613344&{}:{}&-0.175973878972&{}:{}&0.310831265628&, \\ Q_{\mathbf{1b}}&{}\approx{}&1.137649886375&{}:{}&0.130912821404&{}:{}&-0.268562707779&, \\ I^\prime_{\mathbf{1b}}&{}\approx{}&0.791217101774&{}:{}&-0.280216163150&{}:{}&0.488999061376&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{1b}}\)
\(P^-_{\mathbf{1b}}\)
\(P^+_{\mathbf{1b}}\)
\(Q_{\mathbf{1b}}\)
\(I^\prime_{\mathbf{1b}}\)
1b (103)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1b}}&{}\approx{}&0.937152698242&{}:{}&-0.101689070346&{}:{}&0.164536372104&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.427958173665&{}:{}&-0.120409044417&{}:{}&0.692450870753&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.260073510670&{}:{}&-1.260073510670&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.638920185343&{}:{}&-0.409342438319&{}:{}&0.770422252976&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.961266679393&{}:{}&-0.188190960236&{}:{}&0.226924280842&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.839390426438&{}:{}&-0.105283192248&{}:{}&0.265892765810&, \\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.661267438022&{}:{}&-0.186052248131&{}:{}&0.524784810110&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.022908731440&{}:{}&-0.110994332240&{}:{}&0.088085600801&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.873331277549&{}:{}&-0.026662490618&{}:{}&0.153331213069&, \\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.951056516295&{}:{}&1.951056516295&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.309016994375&{}:{}&0.000000000000&{}:{}&-0.309016994375&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.896802246667&{}:{}&0.103197753333&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1b}}}{B^\prime_{\mathbf{1b}}}{C^\prime_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^*_{\mathbf{1b}}}{B^*_{\mathbf{1b}}}{C^*_{\mathbf{1b}}}\)
1b (103)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1b}}}}&{}\approx{}&0.164536372104&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1b}}}}&{}\approx{}&0.692450870753&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1b}}}}&{}\approx{}&2.038841768588&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.937152698242&{}:{}&-0.101689070346&{}:{}&0.164536372104&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.427958173665&{}:{}&-0.120409044417&{}:{}&0.692450870753&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.260073510670&{}:{}&-1.260073510670&{}:{}&1.000000000000&. \end{alignedat} \]
1b (103)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1b}}&{}\approx{}&0.900381905509&{}:{}&-0.112933240919&{}:{}&0.212551335411&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.638920185343&{}:{}&-0.409342438319&{}:{}&0.770422252976&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.961266679393&{}:{}&-0.188190960236&{}:{}&0.226924280842&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.839390426438&{}:{}&-0.105283192248&{}:{}&0.265892765810&. \end{alignedat} \]
1b (103)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1b}}&{}\approx{}&0.947343589942&{}:{}&-0.028922059966&{}:{}&0.081578470024&,\\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.661267438022&{}:{}&-0.186052248131&{}:{}&0.524784810110&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.022908731440&{}:{}&-0.110994332240&{}:{}&0.088085600801&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.873331277549&{}:{}&-0.026662490618&{}:{}&0.153331213069&. \end{alignedat} \]
1b (103)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1b}}&{}\approx{}&0.865142613344&{}:{}&-0.175973878972&{}:{}&0.310831265628&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.937152698242&{}:{}&-0.101689070346&{}:{}&0.164536372104&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.427958173665&{}:{}&-0.120409044417&{}:{}&0.692450870753&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.260073510670&{}:{}&-1.260073510670&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.638920185343&{}:{}&-0.409342438319&{}:{}&0.770422252976&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.961266679393&{}:{}&-0.188190960236&{}:{}&0.226924280842&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.839390426438&{}:{}&-0.105283192248&{}:{}&0.265892765810&, \end{alignedat} \]
1b (103)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1b}}&{}\approx{}&1.137649886375&{}:{}&0.130912821404&{}:{}&-0.268562707779&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.951056516295&{}:{}&1.951056516295&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.309016994375&{}:{}&0.000000000000&{}:{}&-0.309016994375&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.896802246667&{}:{}&0.103197753333&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1b}}&{}\approx{}&0.791217101774&{}:{}&-0.280216163150&{}:{}&0.488999061376&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.638920185343&{}:{}&-0.409342438319&{}:{}&0.770422252976&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.961266679393&{}:{}&-0.188190960236&{}:{}&0.226924280842&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.839390426438&{}:{}&-0.105283192248&{}:{}&0.265892765810&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.951056516295&{}:{}&1.951056516295&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.309016994375&{}:{}&0.000000000000&{}:{}&-0.309016994375&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.896802246667&{}:{}&0.103197753333&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)