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 > 1a (013)

1a(013)

Malfatti circles

1a (013)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{1a}}&{}\approx{}&-0.112933240919&{}:{}&0.900381905509&{}:{}&0.212551335411&, \\ P^-_{\mathbf{1a}}&{}\approx{}&-0.028922059966&{}:{}&0.947343589942&{}:{}&0.081578470024&, \\ P^+_{\mathbf{1a}}&{}\approx{}&-0.175973878972&{}:{}&0.865142613344&{}:{}&0.310831265628&, \\ Q_{\mathbf{1a}}&{}\approx{}&0.130912821404&{}:{}&1.137649886375&{}:{}&-0.268562707779&, \\ I^\prime_{\mathbf{1a}}&{}\approx{}&-0.280216163150&{}:{}&0.791217101774&{}:{}&0.488999061376&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{1a}}\)
\(P^-_{\mathbf{1a}}\)
\(P^+_{\mathbf{1a}}\)
\(Q_{\mathbf{1a}}\)
\(I^\prime_{\mathbf{1a}}\)
1a (013)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1a}}&{}\approx{}&-0.120409044417&{}:{}&0.427958173665&{}:{}&0.692450870753&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.101689070346&{}:{}&0.937152698242&{}:{}&0.164536372104&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-1.260073510670&{}:{}&1.260073510670&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.188190960236&{}:{}&0.961266679393&{}:{}&0.226924280842&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.409342438319&{}:{}&0.638920185343&{}:{}&0.770422252976&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.105283192248&{}:{}&0.839390426438&{}:{}&0.265892765810&, \\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.110994332240&{}:{}&1.022908731440&{}:{}&0.088085600801&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.186052248131&{}:{}&0.661267438022&{}:{}&0.524784810110&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.026662490618&{}:{}&0.873331277549&{}:{}&0.153331213069&, \\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.309016994375&{}:{}&-0.309016994375&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.951056516295&{}:{}&0.000000000000&{}:{}&1.951056516295&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.103197753333&{}:{}&0.896802246667&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1a}}}{B^\prime_{\mathbf{1a}}}{C^\prime_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^*_{\mathbf{1a}}}{B^*_{\mathbf{1a}}}{C^*_{\mathbf{1a}}}\)
1a (013)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1a}}}}&{}\approx{}&0.692450870753&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1a}}}}&{}\approx{}&0.164536372104&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1a}}}}&{}\approx{}&2.038841768588&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.120409044417&{}:{}&0.427958173665&{}:{}&0.692450870753&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.101689070346&{}:{}&0.937152698242&{}:{}&0.164536372104&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-1.260073510670&{}:{}&1.260073510670&{}:{}&1.000000000000&. \end{alignedat} \]
1a (013)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1a}}&{}\approx{}&-0.112933240919&{}:{}&0.900381905509&{}:{}&0.212551335411&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.188190960236&{}:{}&0.961266679393&{}:{}&0.226924280842&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.409342438319&{}:{}&0.638920185343&{}:{}&0.770422252976&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.105283192248&{}:{}&0.839390426438&{}:{}&0.265892765810&. \end{alignedat} \]
1a (013)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1a}}&{}\approx{}&-0.028922059966&{}:{}&0.947343589942&{}:{}&0.081578470024&,\\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.110994332240&{}:{}&1.022908731440&{}:{}&0.088085600801&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.186052248131&{}:{}&0.661267438022&{}:{}&0.524784810110&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.026662490618&{}:{}&0.873331277549&{}:{}&0.153331213069&. \end{alignedat} \]
1a (013)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1a}}&{}\approx{}&-0.175973878972&{}:{}&0.865142613344&{}:{}&0.310831265628&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.120409044417&{}:{}&0.427958173665&{}:{}&0.692450870753&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.101689070346&{}:{}&0.937152698242&{}:{}&0.164536372104&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-1.260073510670&{}:{}&1.260073510670&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.188190960236&{}:{}&0.961266679393&{}:{}&0.226924280842&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.409342438319&{}:{}&0.638920185343&{}:{}&0.770422252976&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.105283192248&{}:{}&0.839390426438&{}:{}&0.265892765810&, \end{alignedat} \]
1a (013)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1a}}&{}\approx{}&0.130912821404&{}:{}&1.137649886375&{}:{}&-0.268562707779&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.309016994375&{}:{}&-0.309016994375&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.951056516295&{}:{}&0.000000000000&{}:{}&1.951056516295&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.103197753333&{}:{}&0.896802246667&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1a}}&{}\approx{}&-0.280216163150&{}:{}&0.791217101774&{}:{}&0.488999061376&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.188190960236&{}:{}&0.961266679393&{}:{}&0.226924280842&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.409342438319&{}:{}&0.638920185343&{}:{}&0.770422252976&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.105283192248&{}:{}&0.839390426438&{}:{}&0.265892765810&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.309016994375&{}:{}&-0.309016994375&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.951056516295&{}:{}&0.000000000000&{}:{}&1.951056516295&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.103197753333&{}:{}&0.896802246667&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)