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 > 0b (101)

0b(101)

Malfatti circles

0b (101)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{b}}&{}\approx{}&0.741300112730&{}:{}&-0.269848945318&{}:{}&0.528548832588&, \\ P^-_{\mathbf{b}}&{}\approx{}&0.796034415883&{}:{}&-0.195627557949&{}:{}&0.399593142066&, \\ P^+_{\mathbf{b}}&{}\approx{}&0.705920518352&{}:{}&-0.317824756245&{}:{}&0.611904237893&, \\ Q_{\mathbf{b}}&{}\approx{}&0.890977102650&{}:{}&-0.106928052948&{}:{}&0.215950950298&, \\ I^\prime_{\mathbf{b}}&{}\approx{}&0.647474008384&{}:{}&-0.390648057027&{}:{}&0.743174048643&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{b}}\)
\(P^-_{\mathbf{b}}\)
\(P^+_{\mathbf{b}}\)
\(Q_{\mathbf{b}}\)
\(I^\prime_{\mathbf{b}}\)
0b (101)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{b}}&{}\approx{}&0.863817568348&{}:{}&-0.160638130892&{}:{}&0.296820562545&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.655202526796&{}:{}&-0.865858935109&{}:{}&1.210656408313&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.370459237183&{}:{}&-0.370459237183&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.473305794955&{}:{}&-0.549392878506&{}:{}&1.076087083552&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.714594158588&{}:{}&-0.224101579146&{}:{}&0.509507420557&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.822741495422&{}:{}&-0.299495333936&{}:{}&0.476753838514&, \\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.420549938664&{}:{}&-0.555762389735&{}:{}&1.135212451070&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.759868775088&{}:{}&-0.141307498512&{}:{}&0.381438723424&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.910861099008&{}:{}&-0.223846518284&{}:{}&0.312985419276&, \\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.980785280403&{}:{}&1.980785280403&,\\B^*_{\mathbf{b}}&{}\approx{}&0.804909677984&{}:{}&0.000000000000&{}:{}&0.195090322016&,\\C^*_{\mathbf{b}}&{}\approx{}&1.136379290286&{}:{}&-0.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{b}}}{B^\prime_{\mathbf{b}}}{C^\prime_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{b}}}{B^{\prime\prime}_{\mathbf{b}}}{C^{\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{b}}}{B^{\prime\prime\prime}_{\mathbf{b}}}{C^{\prime\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^*_{\mathbf{b}}}{B^*_{\mathbf{b}}}{C^*_{\mathbf{b}}}\)
0b (101)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{b}}}}&{}\approx{}&0.296820562545&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{b}}}}&{}\approx{}&1.210656408313&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{b}}}}&{}\approx{}&0.684519413727&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.541196100146&{}:{}&-0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.863817568348&{}:{}&-0.160638130892&{}:{}&0.296820562545&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.655202526796&{}:{}&-0.865858935109&{}:{}&1.210656408313&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.370459237183&{}:{}&-0.370459237183&{}:{}&1.000000000000&. \end{alignedat} \]
0b (101)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{b}}&{}\approx{}&0.741300112730&{}:{}&-0.269848945318&{}:{}&0.528548832588&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.473305794955&{}:{}&-0.549392878506&{}:{}&1.076087083552&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.714594158588&{}:{}&-0.224101579146&{}:{}&0.509507420557&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.822741495422&{}:{}&-0.299495333936&{}:{}&0.476753838514&. \end{alignedat} \]
0b (101)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{b}}&{}\approx{}&0.796034415883&{}:{}&-0.195627557949&{}:{}&0.399593142066&,\\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.420549938664&{}:{}&-0.555762389735&{}:{}&1.135212451070&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.759868775088&{}:{}&-0.141307498512&{}:{}&0.381438723424&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.910861099008&{}:{}&-0.223846518284&{}:{}&0.312985419276&. \end{alignedat} \]
0b (101)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{b}}&{}\approx{}&0.705920518352&{}:{}&-0.317824756245&{}:{}&0.611904237893&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.863817568348&{}:{}&-0.160638130892&{}:{}&0.296820562545&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.655202526796&{}:{}&-0.865858935109&{}:{}&1.210656408313&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.370459237183&{}:{}&-0.370459237183&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.473305794955&{}:{}&-0.549392878506&{}:{}&1.076087083552&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.714594158588&{}:{}&-0.224101579146&{}:{}&0.509507420557&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.822741495422&{}:{}&-0.299495333936&{}:{}&0.476753838514&, \end{alignedat} \]
0b (101)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{b}}&{}\approx{}&0.890977102650&{}:{}&-0.106928052948&{}:{}&0.215950950298&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.980785280403&{}:{}&1.980785280403&,\\B^*_{\mathbf{b}}&{}\approx{}&0.804909677984&{}:{}&0.000000000000&{}:{}&0.195090322016&,\\C^*_{\mathbf{b}}&{}\approx{}&1.136379290286&{}:{}&-0.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{b}}&{}\approx{}&0.647474008384&{}:{}&-0.390648057027&{}:{}&0.743174048643&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.473305794955&{}:{}&-0.549392878506&{}:{}&1.076087083552&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.714594158588&{}:{}&-0.224101579146&{}:{}&0.509507420557&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.822741495422&{}:{}&-0.299495333936&{}:{}&0.476753838514&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-0.980785280403&{}:{}&1.980785280403&,\\B^*_{\mathbf{b}}&{}\approx{}&0.804909677984&{}:{}&0.000000000000&{}:{}&0.195090322016&,\\C^*_{\mathbf{b}}&{}\approx{}&1.136379290286&{}:{}&-0.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)