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 > 5c (312)

5c(312)

Malfatti circles

5c (312)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&, \\ P_{\mathbf{5c}}&{}\approx{}&-0.306012540087&{}:{}&-0.085267471877&{}:{}&1.391280011964&, \\ P^-_{\mathbf{5c}}&{}\approx{}&-0.102414753972&{}:{}&0.102960104927&{}:{}&0.999454649045&, \\ P^+_{\mathbf{5c}}&{}\approx{}&-0.542549947360&{}:{}&-0.303947958753&{}:{}&1.846497906113&, \\ Q_{\mathbf{5c}}&{}\approx{}&-1.048455368038&{}:{}&0.553441891575&{}:{}&1.495013476463&, \\ I^\prime_{\mathbf{5c}}&{}\approx{}&-0.844311517102&{}:{}&-0.445681692658&{}:{}&2.289993209760&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{5c}}\)
\(P^-_{\mathbf{5c}}\)
\(P^+_{\mathbf{5c}}\)
\(Q_{\mathbf{5c}}\)
\(I^\prime_{\mathbf{5c}}\)
5c (312)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5c}}&{}\approx{}&-9.994632805183&{}:{}&-17.789689572610&{}:{}&28.784322377793&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.845417662611&{}:{}&0.477503149817&{}:{}&1.367914512795&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.287128803051&{}:{}&-0.287128803051&{}:{}&1.574257606102&, \\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.428668966623&{}:{}&-0.093275513974&{}:{}&1.521944480596&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.441320981499&{}:{}&-0.565136205662&{}:{}&2.006457187161&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-1.260489493588&{}:{}&-0.351223359719&{}:{}&2.611712853306&, \\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.198115099714&{}:{}&0.111898045574&{}:{}&1.086217054140&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.143287641338&{}:{}&-0.255041151455&{}:{}&1.398328792793&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.346584061832&{}:{}&0.348429596207&{}:{}&0.998154465624&, \\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.270175225787&{}:{}&0.729824774213&,\\B^*_{\mathbf{5c}}&{}\approx{}&-2.347858762963&{}:{}&0.000000000000&{}:{}&3.347858762963&,\\C^*_{\mathbf{5c}}&{}\approx{}&2.118033988750&{}:{}&-1.118033988750&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5c}}}{B^\prime_{\mathbf{5c}}}{C^\prime_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^*_{\mathbf{5c}}}{B^*_{\mathbf{5c}}}{C^*_{\mathbf{5c}}}\)
5c (312)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5c}}}}&{}\approx{}&-6.795056767428&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.322920812428&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.109673443616&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-9.994632805183&{}:{}&-17.789689572610&{}:{}&28.784322377793&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.845417662611&{}:{}&0.477503149817&{}:{}&1.367914512795&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.287128803051&{}:{}&-0.287128803051&{}:{}&1.574257606102&. \end{alignedat} \]
5c (312)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5c}}&{}\approx{}&-0.306012540087&{}:{}&-0.085267471877&{}:{}&1.391280011964&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.428668966623&{}:{}&-0.093275513974&{}:{}&1.521944480596&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.441320981499&{}:{}&-0.565136205662&{}:{}&2.006457187161&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-1.260489493588&{}:{}&-0.351223359719&{}:{}&2.611712853306&. \end{alignedat} \]
5c (312)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5c}}&{}\approx{}&-0.102414753972&{}:{}&0.102960104927&{}:{}&0.999454649045&,\\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.198115099714&{}:{}&0.111898045574&{}:{}&1.086217054140&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.143287641338&{}:{}&-0.255041151455&{}:{}&1.398328792793&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.346584061832&{}:{}&0.348429596207&{}:{}&0.998154465624&. \end{alignedat} \]
5c (312)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5c}}&{}\approx{}&-0.542549947360&{}:{}&-0.303947958753&{}:{}&1.846497906113&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-9.994632805183&{}:{}&-17.789689572610&{}:{}&28.784322377793&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.845417662611&{}:{}&0.477503149817&{}:{}&1.367914512795&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.287128803051&{}:{}&-0.287128803051&{}:{}&1.574257606102&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.428668966623&{}:{}&-0.093275513974&{}:{}&1.521944480596&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.441320981499&{}:{}&-0.565136205662&{}:{}&2.006457187161&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-1.260489493588&{}:{}&-0.351223359719&{}:{}&2.611712853306&, \end{alignedat} \]
5c (312)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5c}}&{}\approx{}&-1.048455368038&{}:{}&0.553441891575&{}:{}&1.495013476463&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.270175225787&{}:{}&0.729824774213&,\\B^*_{\mathbf{5c}}&{}\approx{}&-2.347858762963&{}:{}&0.000000000000&{}:{}&3.347858762963&,\\C^*_{\mathbf{5c}}&{}\approx{}&2.118033988750&{}:{}&-1.118033988750&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5c}}&{}\approx{}&-0.844311517102&{}:{}&-0.445681692658&{}:{}&2.289993209760&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.428668966623&{}:{}&-0.093275513974&{}:{}&1.521944480596&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.441320981499&{}:{}&-0.565136205662&{}:{}&2.006457187161&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-1.260489493588&{}:{}&-0.351223359719&{}:{}&2.611712853306&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.270175225787&{}:{}&0.729824774213&,\\B^*_{\mathbf{5c}}&{}\approx{}&-2.347858762963&{}:{}&0.000000000000&{}:{}&3.347858762963&,\\C^*_{\mathbf{5c}}&{}\approx{}&2.118033988750&{}:{}&-1.118033988750&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)