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 > 4a (211)

4a(211)

Malfatti circles

4a (211)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{4a}}&{}\approx{}&1.125215770933&{}:{}&-0.005645349511&{}:{}&-0.119570421423&, \\ P^-_{\mathbf{4a}}&{}\approx{}&0.685214846521&{}:{}&0.151772991635&{}:{}&0.163012161845&, \\ P^+_{\mathbf{4a}}&{}\approx{}&2.013757169175&{}:{}&-0.323537211072&{}:{}&-0.690219958103&, \\ Q_{\mathbf{4a}}&{}\approx{}&3.395063102620&{}:{}&-0.740115201310&{}:{}&-1.654947901310&, \\ I^\prime_{\mathbf{4a}}&{}\approx{}&1.943587463614&{}:{}&-0.137667555613&{}:{}&-0.805919908001&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{4a}}\)
\(P^-_{\mathbf{4a}}\)
\(P^+_{\mathbf{4a}}\)
\(Q_{\mathbf{4a}}\)
\(I^\prime_{\mathbf{4a}}\)
4a (211)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4a}}&{}\approx{}&2.475319763287&{}:{}&-0.563522005301&{}:{}&-0.911797757986&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&2.817610026505&{}:{}&2.741378763423&{}:{}&-4.558988789928&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.956943567841&{}:{}&-0.956943567841&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.428668966623&{}:{}&-0.019326528304&{}:{}&-0.409342438319&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.912572132841&{}:{}&-0.709333730188&{}:{}&-0.203238402653&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.532368140490&{}:{}&-0.012705210514&{}:{}&-1.519662929976&, \\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.331352024068&{}:{}&0.322387198176&{}:{}&0.346260777755&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.989860760290&{}:{}&-0.225347984887&{}:{}&0.235487224597&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.172138645344&{}:{}&0.259625123008&{}:{}&-0.431763768352&, \\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.309016994375&{}:{}&0.690983005625&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.951056516295&{}:{}&0.000000000000&{}:{}&-0.951056516295&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.278768257918&{}:{}&-0.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4a}}}{B^\prime_{\mathbf{4a}}}{C^\prime_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^*_{\mathbf{4a}}}{B^*_{\mathbf{4a}}}{C^*_{\mathbf{4a}}}\)
4a (211)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4a}}}}&{}\approx{}&-0.911797757986&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4a}}}}&{}\approx{}&-4.558988789928&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4a}}}}&{}\approx{}&-1.548367218082&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&2.475319763287&{}:{}&-0.563522005301&{}:{}&-0.911797757986&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&2.817610026505&{}:{}&2.741378763423&{}:{}&-4.558988789928&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.956943567841&{}:{}&-0.956943567841&{}:{}&1.000000000000&. \end{alignedat} \]
4a (211)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4a}}&{}\approx{}&1.125215770933&{}:{}&-0.005645349511&{}:{}&-0.119570421423&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.428668966623&{}:{}&-0.019326528304&{}:{}&-0.409342438319&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.912572132841&{}:{}&-0.709333730188&{}:{}&-0.203238402653&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.532368140490&{}:{}&-0.012705210514&{}:{}&-1.519662929976&. \end{alignedat} \]
4a (211)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4a}}&{}\approx{}&0.685214846521&{}:{}&0.151772991635&{}:{}&0.163012161845&,\\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.331352024068&{}:{}&0.322387198176&{}:{}&0.346260777755&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.989860760290&{}:{}&-0.225347984887&{}:{}&0.235487224597&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.172138645344&{}:{}&0.259625123008&{}:{}&-0.431763768352&. \end{alignedat} \]
4a (211)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4a}}&{}\approx{}&2.013757169175&{}:{}&-0.323537211072&{}:{}&-0.690219958103&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&2.475319763287&{}:{}&-0.563522005301&{}:{}&-0.911797757986&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&2.817610026505&{}:{}&2.741378763423&{}:{}&-4.558988789928&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.956943567841&{}:{}&-0.956943567841&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.428668966623&{}:{}&-0.019326528304&{}:{}&-0.409342438319&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.912572132841&{}:{}&-0.709333730188&{}:{}&-0.203238402653&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.532368140490&{}:{}&-0.012705210514&{}:{}&-1.519662929976&, \end{alignedat} \]
4a (211)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4a}}&{}\approx{}&3.395063102620&{}:{}&-0.740115201310&{}:{}&-1.654947901310&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.309016994375&{}:{}&0.690983005625&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.951056516295&{}:{}&0.000000000000&{}:{}&-0.951056516295&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.278768257918&{}:{}&-0.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4a}}&{}\approx{}&1.943587463614&{}:{}&-0.137667555613&{}:{}&-0.805919908001&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.428668966623&{}:{}&-0.019326528304&{}:{}&-0.409342438319&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.912572132841&{}:{}&-0.709333730188&{}:{}&-0.203238402653&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.532368140490&{}:{}&-0.012705210514&{}:{}&-1.519662929976&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.309016994375&{}:{}&0.690983005625&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.951056516295&{}:{}&0.000000000000&{}:{}&-0.951056516295&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.278768257918&{}:{}&-0.278768257918&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)