Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 135° Top Angle

\(A=22.5\degree\), \(B=22.5\degree\), \(C=135\degree\).


[Top] > Isosceles Triangle with 135° Top Angle > 2a (031)

2a(031)

Malfatti circles

2a (031)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{2a}}&{}\approx{}&-0.000798135709&{}:{}&0.002192552912&{}:{}&0.998605582797&, \\ P^-_{\mathbf{2a}}&{}\approx{}&0.008019871917&{}:{}&-0.006602701147&{}:{}&0.998582829229&, \\ P^+_{\mathbf{2a}}&{}\approx{}&-0.009337460454&{}:{}&0.010709843190&{}:{}&0.998627617264&, \\ Q_{\mathbf{2a}}&{}\approx{}&0.022877566451&{}:{}&-0.190627130210&{}:{}&1.167749563759&, \\ I^\prime_{\mathbf{2a}}&{}\approx{}&-0.020517324001&{}:{}&0.034006143825&{}:{}&0.986511180176&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{2a}}\)
\(P^-_{\mathbf{2a}}\)
\(P^+_{\mathbf{2a}}\)
\(Q_{\mathbf{2a}}\)
\(I^\prime_{\mathbf{2a}}\)
2a (031)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2a}}&{}\approx{}&0.241791662387&{}:{}&0.266247361627&{}:{}&0.491960975985&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-9.991146350560&{}:{}&-7.470084888655&{}:{}&18.461231239215&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.008843564140&{}:{}&0.008843564140&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.017671486540&{}:{}&0.002229519123&{}:{}&1.015441967417&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.000786196579&{}:{}&0.017118526271&{}:{}&0.983667670309&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.023820911731&{}:{}&0.065438256692&{}:{}&0.958382655039&, \\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.008982392730&{}:{}&-0.006715869615&{}:{}&1.015698262346&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.007897976659&{}:{}&0.008696807106&{}:{}&0.983405216235&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.452211298296&{}:{}&-0.372302212368&{}:{}&0.920090914073&, \\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.195090322016&{}:{}&1.195090322016&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.019214719597&{}:{}&0.000000000000&{}:{}&0.980785280403&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.136379290286&{}:{}&1.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2a}}}{B^\prime_{\mathbf{2a}}}{C^\prime_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^*_{\mathbf{2a}}}{B^*_{\mathbf{2a}}}{C^*_{\mathbf{2a}}}\)
2a (031)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2a}}}}&{}\approx{}&0.491960975985&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2a}}}}&{}\approx{}&18.461231239215&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2a}}}}&{}\approx{}&0.016340775808&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.241791662387&{}:{}&0.266247361627&{}:{}&0.491960975985&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-9.991146350560&{}:{}&-7.470084888655&{}:{}&18.461231239215&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.008843564140&{}:{}&0.008843564140&{}:{}&1.000000000000&. \end{alignedat} \]
2a (031)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2a}}&{}\approx{}&-0.000798135709&{}:{}&0.002192552912&{}:{}&0.998605582797&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.017671486540&{}:{}&0.002229519123&{}:{}&1.015441967417&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.000786196579&{}:{}&0.017118526271&{}:{}&0.983667670309&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.023820911731&{}:{}&0.065438256692&{}:{}&0.958382655039&. \end{alignedat} \]
2a (031)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2a}}&{}\approx{}&0.008019871917&{}:{}&-0.006602701147&{}:{}&0.998582829229&,\\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.008982392730&{}:{}&-0.006715869615&{}:{}&1.015698262346&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.007897976659&{}:{}&0.008696807106&{}:{}&0.983405216235&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.452211298296&{}:{}&-0.372302212368&{}:{}&0.920090914073&. \end{alignedat} \]
2a (031)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2a}}&{}\approx{}&-0.009337460454&{}:{}&0.010709843190&{}:{}&0.998627617264&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.241791662387&{}:{}&0.266247361627&{}:{}&0.491960975985&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-9.991146350560&{}:{}&-7.470084888655&{}:{}&18.461231239215&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.008843564140&{}:{}&0.008843564140&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.017671486540&{}:{}&0.002229519123&{}:{}&1.015441967417&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.000786196579&{}:{}&0.017118526271&{}:{}&0.983667670309&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.023820911731&{}:{}&0.065438256692&{}:{}&0.958382655039&, \end{alignedat} \]
2a (031)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2a}}&{}\approx{}&0.022877566451&{}:{}&-0.190627130210&{}:{}&1.167749563759&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.195090322016&{}:{}&1.195090322016&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.019214719597&{}:{}&0.000000000000&{}:{}&0.980785280403&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.136379290286&{}:{}&1.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2a}}&{}\approx{}&-0.020517324001&{}:{}&0.034006143825&{}:{}&0.986511180176&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.017671486540&{}:{}&0.002229519123&{}:{}&1.015441967417&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.000786196579&{}:{}&0.017118526271&{}:{}&0.983667670309&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.023820911731&{}:{}&0.065438256692&{}:{}&0.958382655039&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.195090322016&{}:{}&1.195090322016&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.019214719597&{}:{}&0.000000000000&{}:{}&0.980785280403&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.136379290286&{}:{}&1.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)