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 > 0a (011)

0a(011)

Malfatti circles

0a (011)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{a}}&{}\approx{}&-0.322205268138&{}:{}&0.715783638318&{}:{}&0.606421629820&, \\ P^-_{\mathbf{a}}&{}\approx{}&-0.241292834512&{}:{}&0.742519250791&{}:{}&0.498773583720&, \\ P^+_{\mathbf{a}}&{}\approx{}&-0.374507320074&{}:{}&0.698501654210&{}:{}&0.676005665864&, \\ Q_{\mathbf{a}}&{}\approx{}&-0.177346747593&{}:{}&0.813526594315&{}:{}&0.363820153278&, \\ I^\prime_{\mathbf{a}}&{}\approx{}&-0.447316704796&{}:{}&0.666714164514&{}:{}&0.780602540282&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{a}}\)
\(P^-_{\mathbf{a}}\)
\(P^+_{\mathbf{a}}\)
\(Q_{\mathbf{a}}\)
\(I^\prime_{\mathbf{a}}\)
0a (011)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{a}}&{}\approx{}&-1.198926561037&{}:{}&0.839915207551&{}:{}&1.359011353486&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.199576037755&{}:{}&0.876655225327&{}:{}&0.322920812428&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.338909579091&{}:{}&0.338909579091&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.251216481318&{}:{}&0.677353438913&{}:{}&0.573863042405&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.586162603175&{}:{}&0.482947614758&{}:{}&1.103214988417&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.391445299440&{}:{}&0.869601363923&{}:{}&0.521843935517&, \\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.157648462322&{}:{}&0.692484678091&{}:{}&0.465163784231&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.565724688594&{}:{}&0.396321830444&{}:{}&1.169402858151&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.311451084882&{}:{}&0.958413981388&{}:{}&0.353037103494&, \\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.690983005625&{}:{}&0.309016994375&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.951056516295&{}:{}&0.000000000000&{}:{}&1.951056516295&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.278768257918&{}:{}&1.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{a}}}{B^\prime_{\mathbf{a}}}{C^\prime_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{a}}}{B^{\prime\prime}_{\mathbf{a}}}{C^{\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{a}}}{B^{\prime\prime\prime}_{\mathbf{a}}}{C^{\prime\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^*_{\mathbf{a}}}{B^*_{\mathbf{a}}}{C^*_{\mathbf{a}}}\)
0a (011)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{a}}}}&{}\approx{}&1.359011353486&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{a}}}}&{}\approx{}&0.322920812428&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{a}}}}&{}\approx{}&0.548367218082&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-1.198926561037&{}:{}&0.839915207551&{}:{}&1.359011353486&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.199576037755&{}:{}&0.876655225327&{}:{}&0.322920812428&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.338909579091&{}:{}&0.338909579091&{}:{}&1.000000000000&. \end{alignedat} \]
0a (011)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{a}}&{}\approx{}&-0.322205268138&{}:{}&0.715783638318&{}:{}&0.606421629820&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.251216481318&{}:{}&0.677353438913&{}:{}&0.573863042405&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.586162603175&{}:{}&0.482947614758&{}:{}&1.103214988417&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.391445299440&{}:{}&0.869601363923&{}:{}&0.521843935517&. \end{alignedat} \]
0a (011)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{a}}&{}\approx{}&-0.241292834512&{}:{}&0.742519250791&{}:{}&0.498773583720&,\\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.157648462322&{}:{}&0.692484678091&{}:{}&0.465163784231&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.565724688594&{}:{}&0.396321830444&{}:{}&1.169402858151&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.311451084882&{}:{}&0.958413981388&{}:{}&0.353037103494&. \end{alignedat} \]
0a (011)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{a}}&{}\approx{}&-0.374507320074&{}:{}&0.698501654210&{}:{}&0.676005665864&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-1.198926561037&{}:{}&0.839915207551&{}:{}&1.359011353486&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.199576037755&{}:{}&0.876655225327&{}:{}&0.322920812428&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.338909579091&{}:{}&0.338909579091&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.251216481318&{}:{}&0.677353438913&{}:{}&0.573863042405&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.586162603175&{}:{}&0.482947614758&{}:{}&1.103214988417&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.391445299440&{}:{}&0.869601363923&{}:{}&0.521843935517&, \end{alignedat} \]
0a (011)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{a}}&{}\approx{}&-0.177346747593&{}:{}&0.813526594315&{}:{}&0.363820153278&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.690983005625&{}:{}&0.309016994375&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.951056516295&{}:{}&0.000000000000&{}:{}&1.951056516295&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.278768257918&{}:{}&1.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{a}}&{}\approx{}&-0.447316704796&{}:{}&0.666714164514&{}:{}&0.780602540282&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.251216481318&{}:{}&0.677353438913&{}:{}&0.573863042405&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.586162603175&{}:{}&0.482947614758&{}:{}&1.103214988417&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.391445299440&{}:{}&0.869601363923&{}:{}&0.521843935517&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.690983005625&{}:{}&0.309016994375&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.951056516295&{}:{}&0.000000000000&{}:{}&1.951056516295&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.278768257918&{}:{}&1.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)