Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 108° Top Angle

\(A=36\degree\), \(B=36\degree\), \(C=108\degree\).

*
[Top] > Isosceles Triangle with 108° Top Angle > 2b (121)

2b(121)

Malfatti circles

2b (121)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{2b}}&{}\approx{}&-0.005645349511&{}:{}&1.125215770933&{}:{}&-0.119570421423&, \\ P^-_{\mathbf{2b}}&{}\approx{}&0.151772991635&{}:{}&0.685214846521&{}:{}&0.163012161845&, \\ P^+_{\mathbf{2b}}&{}\approx{}&-0.323537211072&{}:{}&2.013757169175&{}:{}&-0.690219958103&, \\ Q_{\mathbf{2b}}&{}\approx{}&-0.740115201310&{}:{}&3.395063102620&{}:{}&-1.654947901310&, \\ I^\prime_{\mathbf{2b}}&{}\approx{}&-0.137667555613&{}:{}&1.943587463614&{}:{}&-0.805919908001&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{2b}}\)
\(P^-_{\mathbf{2b}}\)
\(P^+_{\mathbf{2b}}\)
\(Q_{\mathbf{2b}}\)
\(I^\prime_{\mathbf{2b}}\)
2b (121)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2b}}&{}\approx{}&2.741378763423&{}:{}&2.817610026505&{}:{}&-4.558988789928&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.563522005301&{}:{}&2.475319763287&{}:{}&-0.911797757986&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.956943567841&{}:{}&0.956943567841&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.709333730188&{}:{}&1.912572132841&{}:{}&-0.203238402653&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.019326528304&{}:{}&1.428668966623&{}:{}&-0.409342438319&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.012705210514&{}:{}&2.532368140490&{}:{}&-1.519662929976&, \\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.225347984887&{}:{}&0.989860760290&{}:{}&0.235487224597&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.322387198176&{}:{}&0.331352024068&{}:{}&0.346260777755&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.259625123008&{}:{}&1.172138645344&{}:{}&-0.431763768352&, \\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&1.951056516295&{}:{}&-0.951056516295&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.309016994375&{}:{}&0.000000000000&{}:{}&0.690983005625&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.278768257918&{}:{}&1.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2b}}}{B^\prime_{\mathbf{2b}}}{C^\prime_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^*_{\mathbf{2b}}}{B^*_{\mathbf{2b}}}{C^*_{\mathbf{2b}}}\)
2b (121)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2b}}}}&{}\approx{}&-4.558988789928&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2b}}}}&{}\approx{}&-0.911797757986&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.548367218082&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&2.741378763423&{}:{}&2.817610026505&{}:{}&-4.558988789928&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.563522005301&{}:{}&2.475319763287&{}:{}&-0.911797757986&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.956943567841&{}:{}&0.956943567841&{}:{}&1.000000000000&. \end{alignedat} \]
2b (121)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2b}}&{}\approx{}&-0.005645349511&{}:{}&1.125215770933&{}:{}&-0.119570421423&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.709333730188&{}:{}&1.912572132841&{}:{}&-0.203238402653&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.019326528304&{}:{}&1.428668966623&{}:{}&-0.409342438319&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.012705210514&{}:{}&2.532368140490&{}:{}&-1.519662929976&. \end{alignedat} \]
2b (121)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2b}}&{}\approx{}&0.151772991635&{}:{}&0.685214846521&{}:{}&0.163012161845&,\\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.225347984887&{}:{}&0.989860760290&{}:{}&0.235487224597&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.322387198176&{}:{}&0.331352024068&{}:{}&0.346260777755&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.259625123008&{}:{}&1.172138645344&{}:{}&-0.431763768352&. \end{alignedat} \]
2b (121)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2b}}&{}\approx{}&-0.323537211072&{}:{}&2.013757169175&{}:{}&-0.690219958103&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&2.741378763423&{}:{}&2.817610026505&{}:{}&-4.558988789928&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.563522005301&{}:{}&2.475319763287&{}:{}&-0.911797757986&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.956943567841&{}:{}&0.956943567841&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.709333730188&{}:{}&1.912572132841&{}:{}&-0.203238402653&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.019326528304&{}:{}&1.428668966623&{}:{}&-0.409342438319&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.012705210514&{}:{}&2.532368140490&{}:{}&-1.519662929976&, \end{alignedat} \]
2b (121)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2b}}&{}\approx{}&-0.740115201310&{}:{}&3.395063102620&{}:{}&-1.654947901310&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&1.951056516295&{}:{}&-0.951056516295&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.309016994375&{}:{}&0.000000000000&{}:{}&0.690983005625&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.278768257918&{}:{}&1.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2b}}&{}\approx{}&-0.137667555613&{}:{}&1.943587463614&{}:{}&-0.805919908001&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.709333730188&{}:{}&1.912572132841&{}:{}&-0.203238402653&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.019326528304&{}:{}&1.428668966623&{}:{}&-0.409342438319&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.012705210514&{}:{}&2.532368140490&{}:{}&-1.519662929976&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&1.951056516295&{}:{}&-0.951056516295&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.309016994375&{}:{}&0.000000000000&{}:{}&0.690983005625&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.278768257918&{}:{}&1.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)