Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 90° Top Angle

\(A=45\degree\), \(B=45\degree\), \(C=90\degree\).


[Top] > Isosceles Triangle with 90° Top Angle > 4 (200)

4(200)

Malfatti circles

4 (200)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&, \\ P_{\mathbf{4}}&{}\approx{}&0.995641720833&{}:{}&0.001558655842&{}:{}&0.002799623325&, \\ P^-_{\mathbf{4}}&{}\approx{}&1.024460869840&{}:{}&-0.010388739502&{}:{}&-0.014072130338&, \\ P^+_{\mathbf{4}}&{}\approx{}&0.969007100408&{}:{}&0.012600423244&{}:{}&0.018392476348&, \\ Q_{\mathbf{4}}&{}\approx{}&1.138898111994&{}:{}&-0.045061790638&{}:{}&-0.093836321356&, \\ I^\prime_{\mathbf{4}}&{}\approx{}&0.906925509083&{}:{}&0.035883532499&{}:{}&0.057190958418&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{4}}\)
\(P^-_{\mathbf{4}}\)
\(P^+_{\mathbf{4}}\)
\(Q_{\mathbf{4}}\)
\(I^\prime_{\mathbf{4}}\)
4 (200)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4}}&{}\approx{}&0.972473645917&{}:{}&0.011401789184&{}:{}&0.016124564899&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.645452460308&{}:{}&-0.558260083544&{}:{}&0.912807623235&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.207106781187&{}:{}&1.207106781187&{}:{}&-1.414213562373&, \\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.841139138032&{}:{}&0.056813572761&{}:{}&0.102047289207&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.974669147540&{}:{}&0.022590201458&{}:{}&0.002740651002&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.966797155858&{}:{}&0.001513500292&{}:{}&0.031689343850&, \\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.964798685182&{}:{}&0.834466715542&{}:{}&1.130331969639&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.002015664775&{}:{}&0.011748155250&{}:{}&-0.013763820025&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.993600922536&{}:{}&-0.010075798361&{}:{}&0.016474875825&, \\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.324423348821&{}:{}&0.675576651179&,\\B^*_{\mathbf{4}}&{}\approx{}&1.089790213552&{}:{}&0.000000000000&{}:{}&-0.089790213552&,\\C^*_{\mathbf{4}}&{}\approx{}&1.041196100146&{}:{}&-0.041196100146&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4}}}{B^\prime_{\mathbf{4}}}{C^\prime_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4}}}{B^{\prime\prime}_{\mathbf{4}}}{C^{\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4}}}{B^{\prime\prime\prime}_{\mathbf{4}}}{C^{\prime\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^*_{\mathbf{4}}}{B^*_{\mathbf{4}}}{C^*_{\mathbf{4}}}\)
4 (200)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4}}}}&{}\approx{}&0.038928143267&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4}}}}&{}\approx{}&2.203712543852&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4}}}}&{}\approx{}&4.121320343560&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.292893218813&{}:{}&0.292893218813&{}:{}&0.414213562373&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.972473645917&{}:{}&0.011401789184&{}:{}&0.016124564899&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.645452460308&{}:{}&-0.558260083544&{}:{}&0.912807623235&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.207106781187&{}:{}&1.207106781187&{}:{}&-1.414213562373&. \end{alignedat} \]
4 (200)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4}}&{}\approx{}&0.995641720833&{}:{}&0.001558655842&{}:{}&0.002799623325&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.841139138032&{}:{}&0.056813572761&{}:{}&0.102047289207&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.974669147540&{}:{}&0.022590201458&{}:{}&0.002740651002&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.966797155858&{}:{}&0.001513500292&{}:{}&0.031689343850&. \end{alignedat} \]
4 (200)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4}}&{}\approx{}&1.024460869840&{}:{}&-0.010388739502&{}:{}&-0.014072130338&,\\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.964798685182&{}:{}&0.834466715542&{}:{}&1.130331969639&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.002015664775&{}:{}&0.011748155250&{}:{}&-0.013763820025&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.993600922536&{}:{}&-0.010075798361&{}:{}&0.016474875825&. \end{alignedat} \]
4 (200)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4}}&{}\approx{}&0.969007100408&{}:{}&0.012600423244&{}:{}&0.018392476348&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.972473645917&{}:{}&0.011401789184&{}:{}&0.016124564899&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.645452460308&{}:{}&-0.558260083544&{}:{}&0.912807623235&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.207106781187&{}:{}&1.207106781187&{}:{}&-1.414213562373&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.841139138032&{}:{}&0.056813572761&{}:{}&0.102047289207&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.974669147540&{}:{}&0.022590201458&{}:{}&0.002740651002&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.966797155858&{}:{}&0.001513500292&{}:{}&0.031689343850&, \end{alignedat} \]
4 (200)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4}}&{}\approx{}&1.138898111994&{}:{}&-0.045061790638&{}:{}&-0.093836321356&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.324423348821&{}:{}&0.675576651179&,\\B^*_{\mathbf{4}}&{}\approx{}&1.089790213552&{}:{}&0.000000000000&{}:{}&-0.089790213552&,\\C^*_{\mathbf{4}}&{}\approx{}&1.041196100146&{}:{}&-0.041196100146&{}:{}&0.000000000000&. \end{alignedat} \]
4 (200)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4}}&{}\approx{}&0.906925509083&{}:{}&0.035883532499&{}:{}&0.057190958418&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.841139138032&{}:{}&0.056813572761&{}:{}&0.102047289207&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.974669147540&{}:{}&0.022590201458&{}:{}&0.002740651002&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.966797155858&{}:{}&0.001513500292&{}:{}&0.031689343850&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.324423348821&{}:{}&0.675576651179&,\\B^*_{\mathbf{4}}&{}\approx{}&1.089790213552&{}:{}&0.000000000000&{}:{}&-0.089790213552&,\\C^*_{\mathbf{4}}&{}\approx{}&1.041196100146&{}:{}&-0.041196100146&{}:{}&0.000000000000&. \end{alignedat} \]
4 (200)