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 > 4c (310)

4c(310)

Malfatti circles

4c (310)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&, \\ P_{\mathbf{4c}}&{}\approx{}&0.286606649450&{}:{}&1.028589529787&{}:{}&-0.315196179237&, \\ P^-_{\mathbf{4c}}&{}\approx{}&0.123706848795&{}:{}&0.917533018974&{}:{}&-0.041239867769&, \\ P^+_{\mathbf{4c}}&{}\approx{}&0.429533489084&{}:{}&1.126029526936&{}:{}&-0.555563016020&, \\ Q_{\mathbf{4c}}&{}\approx{}&-0.626672900290&{}:{}&1.187186182172&{}:{}&0.439486718118&, \\ I^\prime_{\mathbf{4c}}&{}\approx{}&0.640831647282&{}:{}&1.214008897464&{}:{}&-0.854840544746&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{4c}}\)
\(P^-_{\mathbf{4c}}\)
\(P^+_{\mathbf{4c}}\)
\(Q_{\mathbf{4c}}\)
\(I^\prime_{\mathbf{4c}}\)
4c (310)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4c}}&{}\approx{}&11.612666793933&{}:{}&17.171655583861&{}:{}&-27.784322377793&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.227383673861&{}:{}&1.140530838933&{}:{}&-0.367914512795&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.563522005301&{}:{}&0.563522005301&{}:{}&-0.127044010602&, \\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.324022717861&{}:{}&0.974642045998&{}:{}&-0.298664763859&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.914600318267&{}:{}&1.091233029983&{}:{}&-1.005833348250&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.376175079263&{}:{}&1.350037581612&{}:{}&-0.726212660875&, \\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.172698426755&{}:{}&0.866235812819&{}:{}&-0.038934239574&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.466128157335&{}:{}&0.689263914804&{}:{}&-0.155392072140&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.165991443027&{}:{}&1.231157622459&{}:{}&-0.397149065486&, \\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.729824774213&{}:{}&0.270175225787&,\\B^*_{\mathbf{4c}}&{}\approx{}&3.347858762963&{}:{}&0.000000000000&{}:{}&-2.347858762963&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.118033988750&{}:{}&2.118033988750&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4c}}}{B^\prime_{\mathbf{4c}}}{C^\prime_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^*_{\mathbf{4c}}}{B^*_{\mathbf{4c}}}{C^*_{\mathbf{4c}}}\)
4c (310)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4c}}}}&{}\approx{}&6.558988789928&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4c}}}}&{}\approx{}&0.086852834928&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4c}}}}&{}\approx{}&0.215246252616&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.618033988750&{}:{}&2.618033988750&{}:{}&-4.236067977500&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&11.612666793933&{}:{}&17.171655583861&{}:{}&-27.784322377793&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.227383673861&{}:{}&1.140530838933&{}:{}&-0.367914512795&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.563522005301&{}:{}&0.563522005301&{}:{}&-0.127044010602&. \end{alignedat} \]
4c (310)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4c}}&{}\approx{}&0.286606649450&{}:{}&1.028589529787&{}:{}&-0.315196179237&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.324022717861&{}:{}&0.974642045998&{}:{}&-0.298664763859&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.914600318267&{}:{}&1.091233029983&{}:{}&-1.005833348250&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.376175079263&{}:{}&1.350037581612&{}:{}&-0.726212660875&. \end{alignedat} \]
4c (310)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4c}}&{}\approx{}&0.123706848795&{}:{}&0.917533018974&{}:{}&-0.041239867769&,\\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.172698426755&{}:{}&0.866235812819&{}:{}&-0.038934239574&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.466128157335&{}:{}&0.689263914804&{}:{}&-0.155392072140&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.165991443027&{}:{}&1.231157622459&{}:{}&-0.397149065486&. \end{alignedat} \]
4c (310)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4c}}&{}\approx{}&0.429533489084&{}:{}&1.126029526936&{}:{}&-0.555563016020&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&11.612666793933&{}:{}&17.171655583861&{}:{}&-27.784322377793&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.227383673861&{}:{}&1.140530838933&{}:{}&-0.367914512795&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.563522005301&{}:{}&0.563522005301&{}:{}&-0.127044010602&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.324022717861&{}:{}&0.974642045998&{}:{}&-0.298664763859&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.914600318267&{}:{}&1.091233029983&{}:{}&-1.005833348250&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.376175079263&{}:{}&1.350037581612&{}:{}&-0.726212660875&, \end{alignedat} \]
4c (310)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4c}}&{}\approx{}&-0.626672900290&{}:{}&1.187186182172&{}:{}&0.439486718118&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.729824774213&{}:{}&0.270175225787&,\\B^*_{\mathbf{4c}}&{}\approx{}&3.347858762963&{}:{}&0.000000000000&{}:{}&-2.347858762963&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.118033988750&{}:{}&2.118033988750&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4c}}&{}\approx{}&0.640831647282&{}:{}&1.214008897464&{}:{}&-0.854840544746&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.324022717861&{}:{}&0.974642045998&{}:{}&-0.298664763859&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.914600318267&{}:{}&1.091233029983&{}:{}&-1.005833348250&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.376175079263&{}:{}&1.350037581612&{}:{}&-0.726212660875&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.729824774213&{}:{}&0.270175225787&,\\B^*_{\mathbf{4c}}&{}\approx{}&3.347858762963&{}:{}&0.000000000000&{}:{}&-2.347858762963&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.118033988750&{}:{}&2.118033988750&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)