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 > 0a (011)

0a(011)

Malfatti circles

0a (011)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&, \\ P_{\mathbf{a}}&{}\approx{}&-0.269848945318&{}:{}&0.741300112730&{}:{}&0.528548832588&, \\ P^-_{\mathbf{a}}&{}\approx{}&-0.195627557949&{}:{}&0.796034415883&{}:{}&0.399593142066&, \\ P^+_{\mathbf{a}}&{}\approx{}&-0.317824756245&{}:{}&0.705920518352&{}:{}&0.611904237893&, \\ Q_{\mathbf{a}}&{}\approx{}&-0.106928052948&{}:{}&0.890977102650&{}:{}&0.215950950298&, \\ I^\prime_{\mathbf{a}}&{}\approx{}&-0.390648057027&{}:{}&0.647474008384&{}:{}&0.743174048643&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{a}}\)
\(P^-_{\mathbf{a}}\)
\(P^+_{\mathbf{a}}\)
\(Q_{\mathbf{a}}\)
\(I^\prime_{\mathbf{a}}\)
0a (011)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{a}}&{}\approx{}&-0.865858935109&{}:{}&0.655202526796&{}:{}&1.210656408313&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.160638130892&{}:{}&0.863817568348&{}:{}&0.296820562545&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.370459237183&{}:{}&0.370459237183&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.224101579146&{}:{}&0.714594158588&{}:{}&0.509507420557&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.549392878506&{}:{}&0.473305794955&{}:{}&1.076087083552&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.299495333936&{}:{}&0.822741495422&{}:{}&0.476753838514&, \\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.141307498512&{}:{}&0.759868775088&{}:{}&0.381438723424&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.555762389735&{}:{}&0.420549938664&{}:{}&1.135212451070&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.223846518284&{}:{}&0.910861099008&{}:{}&0.312985419276&, \\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.804909677984&{}:{}&0.195090322016&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.980785280403&{}:{}&0.000000000000&{}:{}&1.980785280403&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.136379290286&{}:{}&1.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{a}}}{B^\prime_{\mathbf{a}}}{C^\prime_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{a}}}{B^{\prime\prime}_{\mathbf{a}}}{C^{\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{a}}}{B^{\prime\prime\prime}_{\mathbf{a}}}{C^{\prime\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^*_{\mathbf{a}}}{B^*_{\mathbf{a}}}{C^*_{\mathbf{a}}}\)
0a (011)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{a}}}}&{}\approx{}&1.210656408313&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{a}}}}&{}\approx{}&0.296820562545&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{a}}}}&{}\approx{}&0.684519413727&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.541196100146&{}:{}&0.541196100146&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-0.865858935109&{}:{}&0.655202526796&{}:{}&1.210656408313&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.160638130892&{}:{}&0.863817568348&{}:{}&0.296820562545&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.370459237183&{}:{}&0.370459237183&{}:{}&1.000000000000&. \end{alignedat} \]
0a (011)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{a}}&{}\approx{}&-0.269848945318&{}:{}&0.741300112730&{}:{}&0.528548832588&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.224101579146&{}:{}&0.714594158588&{}:{}&0.509507420557&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.549392878506&{}:{}&0.473305794955&{}:{}&1.076087083552&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.299495333936&{}:{}&0.822741495422&{}:{}&0.476753838514&. \end{alignedat} \]
0a (011)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{a}}&{}\approx{}&-0.195627557949&{}:{}&0.796034415883&{}:{}&0.399593142066&,\\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.141307498512&{}:{}&0.759868775088&{}:{}&0.381438723424&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.555762389735&{}:{}&0.420549938664&{}:{}&1.135212451070&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.223846518284&{}:{}&0.910861099008&{}:{}&0.312985419276&. \end{alignedat} \]
0a (011)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{a}}&{}\approx{}&-0.317824756245&{}:{}&0.705920518352&{}:{}&0.611904237893&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-0.865858935109&{}:{}&0.655202526796&{}:{}&1.210656408313&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.160638130892&{}:{}&0.863817568348&{}:{}&0.296820562545&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.370459237183&{}:{}&0.370459237183&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.224101579146&{}:{}&0.714594158588&{}:{}&0.509507420557&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.549392878506&{}:{}&0.473305794955&{}:{}&1.076087083552&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.299495333936&{}:{}&0.822741495422&{}:{}&0.476753838514&, \end{alignedat} \]
0a (011)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{a}}&{}\approx{}&-0.106928052948&{}:{}&0.890977102650&{}:{}&0.215950950298&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.804909677984&{}:{}&0.195090322016&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.980785280403&{}:{}&0.000000000000&{}:{}&1.980785280403&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.136379290286&{}:{}&1.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{a}}&{}\approx{}&-0.390648057027&{}:{}&0.647474008384&{}:{}&0.743174048643&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.224101579146&{}:{}&0.714594158588&{}:{}&0.509507420557&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.549392878506&{}:{}&0.473305794955&{}:{}&1.076087083552&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.299495333936&{}:{}&0.822741495422&{}:{}&0.476753838514&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.804909677984&{}:{}&0.195090322016&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.980785280403&{}:{}&0.000000000000&{}:{}&1.980785280403&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.136379290286&{}:{}&1.136379290286&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)