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 > 7c (332)

7c(332)

Malfatti circles

7c (332)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&, \\ P_{\mathbf{7c}}&{}\approx{}&-0.203038350841&{}:{}&-0.203038350841&{}:{}&1.406076701682&, \\ P^-_{\mathbf{7c}}&{}\approx{}&-0.037643518691&{}:{}&-0.037643518691&{}:{}&1.075287037382&, \\ P^+_{\mathbf{7c}}&{}\approx{}&-0.376927630039&{}:{}&-0.376927630039&{}:{}&1.753855260078&, \\ Q_{\mathbf{7c}}&{}\approx{}&0.261532306397&{}:{}&0.261532306397&{}:{}&0.476935387207&, \\ I^\prime_{\mathbf{7c}}&{}\approx{}&-0.634051832675&{}:{}&-0.634051832675&{}:{}&2.268103665351&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{7c}}\)
\(P^-_{\mathbf{7c}}\)
\(P^+_{\mathbf{7c}}\)
\(Q_{\mathbf{7c}}\)
\(I^\prime_{\mathbf{7c}}\)
7c (332)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7c}}&{}\approx{}&-0.382683432365&{}:{}&-1.630986313698&{}:{}&3.013669746063&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.630986313698&{}:{}&-0.382683432365&{}:{}&3.013669746063&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.212662112836&{}:{}&-0.212662112836&{}:{}&1.425324225672&, \\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.376263706774&{}:{}&-0.232273819991&{}:{}&1.608537526765&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.232273819991&{}:{}&-0.376263706774&{}:{}&1.608537526765&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-1.006834873031&{}:{}&-1.006834873031&{}:{}&3.013669746063&, \\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.182893532196&{}:{}&-0.042912882880&{}:{}&1.225806415076&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.042912882880&{}:{}&-0.182893532196&{}:{}&1.225806415076&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.170209910570&{}:{}&-0.170209910570&{}:{}&1.340419821139&, \\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.354155379662&{}:{}&0.645844620338&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.354155379662&{}:{}&0.000000000000&{}:{}&0.645844620338&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7c}}}{B^\prime_{\mathbf{7c}}}{C^\prime_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^*_{\mathbf{7c}}}{B^*_{\mathbf{7c}}}{C^*_{\mathbf{7c}}}\)
7c (332)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.248302881333&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.248302881333&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.032375878892&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&6.568535592272&{}:{}&6.568535592272&{}:{}&-12.137071184544&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&-0.382683432365&{}:{}&-1.630986313698&{}:{}&3.013669746063&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.630986313698&{}:{}&-0.382683432365&{}:{}&3.013669746063&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.212662112836&{}:{}&-0.212662112836&{}:{}&1.425324225672&. \end{alignedat} \]
7c (332)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7c}}&{}\approx{}&-0.203038350841&{}:{}&-0.203038350841&{}:{}&1.406076701682&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.376263706774&{}:{}&-0.232273819991&{}:{}&1.608537526765&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.232273819991&{}:{}&-0.376263706774&{}:{}&1.608537526765&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-1.006834873031&{}:{}&-1.006834873031&{}:{}&3.013669746063&. \end{alignedat} \]
7c (332)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7c}}&{}\approx{}&-0.037643518691&{}:{}&-0.037643518691&{}:{}&1.075287037382&,\\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.182893532196&{}:{}&-0.042912882880&{}:{}&1.225806415076&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.042912882880&{}:{}&-0.182893532196&{}:{}&1.225806415076&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.170209910570&{}:{}&-0.170209910570&{}:{}&1.340419821139&. \end{alignedat} \]
7c (332)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7c}}&{}\approx{}&-0.376927630039&{}:{}&-0.376927630039&{}:{}&1.753855260078&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&-0.382683432365&{}:{}&-1.630986313698&{}:{}&3.013669746063&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.630986313698&{}:{}&-0.382683432365&{}:{}&3.013669746063&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.212662112836&{}:{}&-0.212662112836&{}:{}&1.425324225672&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.376263706774&{}:{}&-0.232273819991&{}:{}&1.608537526765&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.232273819991&{}:{}&-0.376263706774&{}:{}&1.608537526765&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-1.006834873031&{}:{}&-1.006834873031&{}:{}&3.013669746063&, \end{alignedat} \]
7c (332)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7c}}&{}\approx{}&0.261532306397&{}:{}&0.261532306397&{}:{}&0.476935387207&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.354155379662&{}:{}&0.645844620338&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.354155379662&{}:{}&0.000000000000&{}:{}&0.645844620338&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7c}}&{}\approx{}&-0.634051832675&{}:{}&-0.634051832675&{}:{}&2.268103665351&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.376263706774&{}:{}&-0.232273819991&{}:{}&1.608537526765&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.232273819991&{}:{}&-0.376263706774&{}:{}&1.608537526765&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-1.006834873031&{}:{}&-1.006834873031&{}:{}&3.013669746063&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.354155379662&{}:{}&0.645844620338&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.354155379662&{}:{}&0.000000000000&{}:{}&0.645844620338&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)