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 > 3c (132)

3c(132)

Malfatti circles

3c (132)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&, \\ P_{\mathbf{3c}}&{}\approx{}&-0.085267471877&{}:{}&-0.306012540087&{}:{}&1.391280011964&, \\ P^-_{\mathbf{3c}}&{}\approx{}&0.102960104927&{}:{}&-0.102414753972&{}:{}&0.999454649045&, \\ P^+_{\mathbf{3c}}&{}\approx{}&-0.303947958753&{}:{}&-0.542549947360&{}:{}&1.846497906113&, \\ Q_{\mathbf{3c}}&{}\approx{}&0.553441891575&{}:{}&-1.048455368038&{}:{}&1.495013476463&, \\ I^\prime_{\mathbf{3c}}&{}\approx{}&-0.445681692658&{}:{}&-0.844311517102&{}:{}&2.289993209760&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{3c}}\)
\(P^-_{\mathbf{3c}}\)
\(P^+_{\mathbf{3c}}\)
\(Q_{\mathbf{3c}}\)
\(I^\prime_{\mathbf{3c}}\)
3c (132)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3c}}&{}\approx{}&0.477503149817&{}:{}&-0.845417662611&{}:{}&1.367914512795&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-17.789689572610&{}:{}&-9.994632805183&{}:{}&28.784322377793&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.287128803051&{}:{}&-0.287128803051&{}:{}&1.574257606102&, \\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.565136205662&{}:{}&-0.441320981499&{}:{}&2.006457187161&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.093275513974&{}:{}&-0.428668966623&{}:{}&1.521944480596&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.351223359719&{}:{}&-1.260489493588&{}:{}&2.611712853306&, \\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.255041151455&{}:{}&-0.143287641338&{}:{}&1.398328792793&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.111898045574&{}:{}&-0.198115099714&{}:{}&1.086217054140&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.348429596207&{}:{}&-0.346584061832&{}:{}&0.998154465624&, \\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-2.347858762963&{}:{}&3.347858762963&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.270175225787&{}:{}&0.000000000000&{}:{}&0.729824774213&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.118033988750&{}:{}&2.118033988750&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3c}}}{B^\prime_{\mathbf{3c}}}{C^\prime_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^*_{\mathbf{3c}}}{B^*_{\mathbf{3c}}}{C^*_{\mathbf{3c}}}\)
3c (132)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.322920812428&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3c}}}}&{}\approx{}&-6.795056767428&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.109673443616&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.477503149817&{}:{}&-0.845417662611&{}:{}&1.367914512795&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-17.789689572610&{}:{}&-9.994632805183&{}:{}&28.784322377793&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.287128803051&{}:{}&-0.287128803051&{}:{}&1.574257606102&. \end{alignedat} \]
3c (132)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3c}}&{}\approx{}&-0.085267471877&{}:{}&-0.306012540087&{}:{}&1.391280011964&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.565136205662&{}:{}&-0.441320981499&{}:{}&2.006457187161&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.093275513974&{}:{}&-0.428668966623&{}:{}&1.521944480596&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.351223359719&{}:{}&-1.260489493588&{}:{}&2.611712853306&. \end{alignedat} \]
3c (132)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3c}}&{}\approx{}&0.102960104927&{}:{}&-0.102414753972&{}:{}&0.999454649045&,\\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.255041151455&{}:{}&-0.143287641338&{}:{}&1.398328792793&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.111898045574&{}:{}&-0.198115099714&{}:{}&1.086217054140&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.348429596207&{}:{}&-0.346584061832&{}:{}&0.998154465624&. \end{alignedat} \]
3c (132)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3c}}&{}\approx{}&-0.303947958753&{}:{}&-0.542549947360&{}:{}&1.846497906113&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.477503149817&{}:{}&-0.845417662611&{}:{}&1.367914512795&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-17.789689572610&{}:{}&-9.994632805183&{}:{}&28.784322377793&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.287128803051&{}:{}&-0.287128803051&{}:{}&1.574257606102&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.565136205662&{}:{}&-0.441320981499&{}:{}&2.006457187161&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.093275513974&{}:{}&-0.428668966623&{}:{}&1.521944480596&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.351223359719&{}:{}&-1.260489493588&{}:{}&2.611712853306&, \end{alignedat} \]
3c (132)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3c}}&{}\approx{}&0.553441891575&{}:{}&-1.048455368038&{}:{}&1.495013476463&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-2.347858762963&{}:{}&3.347858762963&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.270175225787&{}:{}&0.000000000000&{}:{}&0.729824774213&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.118033988750&{}:{}&2.118033988750&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3c}}&{}\approx{}&-0.445681692658&{}:{}&-0.844311517102&{}:{}&2.289993209760&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.565136205662&{}:{}&-0.441320981499&{}:{}&2.006457187161&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.093275513974&{}:{}&-0.428668966623&{}:{}&1.521944480596&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.351223359719&{}:{}&-1.260489493588&{}:{}&2.611712853306&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-2.347858762963&{}:{}&3.347858762963&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.270175225787&{}:{}&0.000000000000&{}:{}&0.729824774213&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.118033988750&{}:{}&2.118033988750&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)