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 > 4b (301)

4b(301)

Malfatti circles

4b (301)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{4b}}&{}\approx{}&0.002192552912&{}:{}&-0.000798135709&{}:{}&0.998605582797&, \\ P^-_{\mathbf{4b}}&{}\approx{}&-0.006602701147&{}:{}&0.008019871917&{}:{}&0.998582829229&, \\ P^+_{\mathbf{4b}}&{}\approx{}&0.010709843190&{}:{}&-0.009337460454&{}:{}&0.998627617264&, \\ Q_{\mathbf{4b}}&{}\approx{}&-0.190627130210&{}:{}&0.022877566451&{}:{}&1.167749563759&, \\ I^\prime_{\mathbf{4b}}&{}\approx{}&0.034006143825&{}:{}&-0.020517324001&{}:{}&0.986511180176&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{4b}}\)
\(P^-_{\mathbf{4b}}\)
\(P^+_{\mathbf{4b}}\)
\(Q_{\mathbf{4b}}\)
\(I^\prime_{\mathbf{4b}}\)
4b (301)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4b}}&{}\approx{}&-7.470084888655&{}:{}&-9.991146350560&{}:{}&18.461231239215&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.266247361627&{}:{}&0.241791662387&{}:{}&0.491960975985&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.008843564140&{}:{}&-0.008843564140&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.017118526271&{}:{}&-0.000786196579&{}:{}&0.983667670309&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.002229519123&{}:{}&-0.017671486540&{}:{}&1.015441967417&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.065438256692&{}:{}&-0.023820911731&{}:{}&0.958382655039&, \\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.008696807106&{}:{}&0.007897976659&{}:{}&0.983405216235&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.006715869615&{}:{}&-0.008982392730&{}:{}&1.015698262346&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.372302212368&{}:{}&0.452211298296&{}:{}&0.920090914073&, \\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.019214719597&{}:{}&0.980785280403&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.195090322016&{}:{}&0.000000000000&{}:{}&1.195090322016&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.136379290286&{}:{}&-0.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4b}}}{B^\prime_{\mathbf{4b}}}{C^\prime_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^*_{\mathbf{4b}}}{B^*_{\mathbf{4b}}}{C^*_{\mathbf{4b}}}\)
4b (301)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4b}}}}&{}\approx{}&18.461231239215&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4b}}}}&{}\approx{}&0.491960975985&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4b}}}}&{}\approx{}&0.016340775808&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-7.470084888655&{}:{}&-9.991146350560&{}:{}&18.461231239215&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.266247361627&{}:{}&0.241791662387&{}:{}&0.491960975985&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.008843564140&{}:{}&-0.008843564140&{}:{}&1.000000000000&. \end{alignedat} \]
4b (301)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4b}}&{}\approx{}&0.002192552912&{}:{}&-0.000798135709&{}:{}&0.998605582797&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.017118526271&{}:{}&-0.000786196579&{}:{}&0.983667670309&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.002229519123&{}:{}&-0.017671486540&{}:{}&1.015441967417&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.065438256692&{}:{}&-0.023820911731&{}:{}&0.958382655039&. \end{alignedat} \]
4b (301)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4b}}&{}\approx{}&-0.006602701147&{}:{}&0.008019871917&{}:{}&0.998582829229&,\\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.008696807106&{}:{}&0.007897976659&{}:{}&0.983405216235&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.006715869615&{}:{}&-0.008982392730&{}:{}&1.015698262346&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.372302212368&{}:{}&0.452211298296&{}:{}&0.920090914073&. \end{alignedat} \]
4b (301)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4b}}&{}\approx{}&0.010709843190&{}:{}&-0.009337460454&{}:{}&0.998627617264&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-7.470084888655&{}:{}&-9.991146350560&{}:{}&18.461231239215&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.266247361627&{}:{}&0.241791662387&{}:{}&0.491960975985&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.008843564140&{}:{}&-0.008843564140&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.017118526271&{}:{}&-0.000786196579&{}:{}&0.983667670309&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.002229519123&{}:{}&-0.017671486540&{}:{}&1.015441967417&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.065438256692&{}:{}&-0.023820911731&{}:{}&0.958382655039&, \end{alignedat} \]
4b (301)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4b}}&{}\approx{}&-0.190627130210&{}:{}&0.022877566451&{}:{}&1.167749563759&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.019214719597&{}:{}&0.980785280403&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.195090322016&{}:{}&0.000000000000&{}:{}&1.195090322016&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.136379290286&{}:{}&-0.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4b}}&{}\approx{}&0.034006143825&{}:{}&-0.020517324001&{}:{}&0.986511180176&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.017118526271&{}:{}&-0.000786196579&{}:{}&0.983667670309&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.002229519123&{}:{}&-0.017671486540&{}:{}&1.015441967417&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.065438256692&{}:{}&-0.023820911731&{}:{}&0.958382655039&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.019214719597&{}:{}&0.980785280403&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.195090322016&{}:{}&0.000000000000&{}:{}&1.195090322016&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.136379290286&{}:{}&-0.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)