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 > 0b (101)

0b(101)

Malfatti circles

0b (101)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{b}}&{}\approx{}&0.715783638318&{}:{}&-0.322205268138&{}:{}&0.606421629820&, \\ P^-_{\mathbf{b}}&{}\approx{}&0.742519250791&{}:{}&-0.241292834512&{}:{}&0.498773583720&, \\ P^+_{\mathbf{b}}&{}\approx{}&0.698501654210&{}:{}&-0.374507320074&{}:{}&0.676005665864&, \\ Q_{\mathbf{b}}&{}\approx{}&0.813526594315&{}:{}&-0.177346747593&{}:{}&0.363820153278&, \\ I^\prime_{\mathbf{b}}&{}\approx{}&0.666714164514&{}:{}&-0.447316704796&{}:{}&0.780602540282&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{b}}\)
\(P^-_{\mathbf{b}}\)
\(P^+_{\mathbf{b}}\)
\(Q_{\mathbf{b}}\)
\(I^\prime_{\mathbf{b}}\)
0b (101)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{b}}&{}\approx{}&0.876655225327&{}:{}&-0.199576037755&{}:{}&0.322920812428&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.839915207551&{}:{}&-1.198926561037&{}:{}&1.359011353486&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.338909579091&{}:{}&-0.338909579091&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.482947614758&{}:{}&-0.586162603175&{}:{}&1.103214988417&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.677353438913&{}:{}&-0.251216481318&{}:{}&0.573863042405&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.869601363923&{}:{}&-0.391445299440&{}:{}&0.521843935517&, \\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.396321830444&{}:{}&-0.565724688594&{}:{}&1.169402858151&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.692484678091&{}:{}&-0.157648462322&{}:{}&0.465163784231&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.958413981388&{}:{}&-0.311451084882&{}:{}&0.353037103494&, \\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.951056516295&{}:{}&1.951056516295&,\\B^*_{\mathbf{b}}&{}\approx{}&0.690983005625&{}:{}&0.000000000000&{}:{}&0.309016994375&,\\C^*_{\mathbf{b}}&{}\approx{}&1.278768257918&{}:{}&-0.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{b}}}{B^\prime_{\mathbf{b}}}{C^\prime_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{b}}}{B^{\prime\prime}_{\mathbf{b}}}{C^{\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{b}}}{B^{\prime\prime\prime}_{\mathbf{b}}}{C^{\prime\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^*_{\mathbf{b}}}{B^*_{\mathbf{b}}}{C^*_{\mathbf{b}}}\)
0b (101)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{b}}}}&{}\approx{}&0.322920812428&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{b}}}}&{}\approx{}&1.359011353486&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{b}}}}&{}\approx{}&0.548367218082&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.876655225327&{}:{}&-0.199576037755&{}:{}&0.322920812428&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.839915207551&{}:{}&-1.198926561037&{}:{}&1.359011353486&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.338909579091&{}:{}&-0.338909579091&{}:{}&1.000000000000&. \end{alignedat} \]
0b (101)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{b}}&{}\approx{}&0.715783638318&{}:{}&-0.322205268138&{}:{}&0.606421629820&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.482947614758&{}:{}&-0.586162603175&{}:{}&1.103214988417&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.677353438913&{}:{}&-0.251216481318&{}:{}&0.573863042405&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.869601363923&{}:{}&-0.391445299440&{}:{}&0.521843935517&. \end{alignedat} \]
0b (101)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{b}}&{}\approx{}&0.742519250791&{}:{}&-0.241292834512&{}:{}&0.498773583720&,\\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.396321830444&{}:{}&-0.565724688594&{}:{}&1.169402858151&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.692484678091&{}:{}&-0.157648462322&{}:{}&0.465163784231&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.958413981388&{}:{}&-0.311451084882&{}:{}&0.353037103494&. \end{alignedat} \]
0b (101)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{b}}&{}\approx{}&0.698501654210&{}:{}&-0.374507320074&{}:{}&0.676005665864&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.876655225327&{}:{}&-0.199576037755&{}:{}&0.322920812428&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.839915207551&{}:{}&-1.198926561037&{}:{}&1.359011353486&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.338909579091&{}:{}&-0.338909579091&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.482947614758&{}:{}&-0.586162603175&{}:{}&1.103214988417&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.677353438913&{}:{}&-0.251216481318&{}:{}&0.573863042405&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.869601363923&{}:{}&-0.391445299440&{}:{}&0.521843935517&, \end{alignedat} \]
0b (101)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{b}}&{}\approx{}&0.813526594315&{}:{}&-0.177346747593&{}:{}&0.363820153278&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.951056516295&{}:{}&1.951056516295&,\\B^*_{\mathbf{b}}&{}\approx{}&0.690983005625&{}:{}&0.000000000000&{}:{}&0.309016994375&,\\C^*_{\mathbf{b}}&{}\approx{}&1.278768257918&{}:{}&-0.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{b}}&{}\approx{}&0.666714164514&{}:{}&-0.447316704796&{}:{}&0.780602540282&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.482947614758&{}:{}&-0.586162603175&{}:{}&1.103214988417&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.677353438913&{}:{}&-0.251216481318&{}:{}&0.573863042405&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.869601363923&{}:{}&-0.391445299440&{}:{}&0.521843935517&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.951056516295&{}:{}&1.951056516295&,\\B^*_{\mathbf{b}}&{}\approx{}&0.690983005625&{}:{}&0.000000000000&{}:{}&0.309016994375&,\\C^*_{\mathbf{b}}&{}\approx{}&1.278768257918&{}:{}&-0.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)