Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 108° Top Angle

\(A=36\degree\), \(B=36\degree\), \(C=108\degree\).

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[Top] > Isosceles Triangle with 108° Top Angle > 3b (123)

3b(123)

Malfatti circles

3b (123)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{3b}}&{}\approx{}&-0.001566642846&{}:{}&1.120652109335&{}:{}&-0.119085466488&, \\ P^-_{\mathbf{3b}}&{}\approx{}&0.091831131859&{}:{}&0.858564880576&{}:{}&0.049603987565&, \\ P^+_{\mathbf{3b}}&{}\approx{}&-0.135274222954&{}:{}&1.495854294235&{}:{}&-0.360580071281&, \\ Q_{\mathbf{3b}}&{}\approx{}&0.183349467982&{}:{}&1.593331342029&{}:{}&-0.776680810011&, \\ I^\prime_{\mathbf{3b}}&{}\approx{}&-0.068234650416&{}:{}&1.824968519586&{}:{}&-0.756733869169&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{3b}}\)
\(P^-_{\mathbf{3b}}\)
\(P^+_{\mathbf{3b}}\)
\(Q_{\mathbf{3b}}\)
\(I^\prime_{\mathbf{3b}}\)
3b (123)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3b}}&{}\approx{}&2.680881290508&{}:{}&2.719723059095&{}:{}&-4.400604349603&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.151564971415&{}:{}&1.396802246667&{}:{}&-0.245237275253&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.878107499420&{}:{}&1.878107499420&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.280493836866&{}:{}&1.432743521885&{}:{}&-0.152249685018&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.015865964599&{}:{}&2.221888000710&{}:{}&-1.206022036111&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002050317308&{}:{}&1.466634479794&{}:{}&-0.464584162486&, \\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.114307721041&{}:{}&1.053444473822&{}:{}&0.060863247219&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.391442488392&{}:{}&0.397113876604&{}:{}&0.211443635003&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.114838102404&{}:{}&1.073665974494&{}:{}&-0.188504076898&, \\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.951056516295&{}:{}&-0.951056516295&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.309016994375&{}:{}&0.000000000000&{}:{}&1.309016994375&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.103197753333&{}:{}&0.896802246667&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3b}}}{B^\prime_{\mathbf{3b}}}{C^\prime_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^*_{\mathbf{3b}}}{B^*_{\mathbf{3b}}}{C^*_{\mathbf{3b}}}\)
3b (123)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3b}}}}&{}\approx{}&-4.400604349603&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3b}}}}&{}\approx{}&-0.245237275253&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3b}}}}&{}\approx{}&-3.038841768588&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.618033988750&{}:{}&-0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&2.680881290508&{}:{}&2.719723059095&{}:{}&-4.400604349603&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.151564971415&{}:{}&1.396802246667&{}:{}&-0.245237275253&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.878107499420&{}:{}&1.878107499420&{}:{}&1.000000000000&. \end{alignedat} \]
3b (123)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3b}}&{}\approx{}&-0.001566642846&{}:{}&1.120652109335&{}:{}&-0.119085466488&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.280493836866&{}:{}&1.432743521885&{}:{}&-0.152249685018&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.015865964599&{}:{}&2.221888000710&{}:{}&-1.206022036111&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002050317308&{}:{}&1.466634479794&{}:{}&-0.464584162486&. \end{alignedat} \]
3b (123)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3b}}&{}\approx{}&0.091831131859&{}:{}&0.858564880576&{}:{}&0.049603987565&,\\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.114307721041&{}:{}&1.053444473822&{}:{}&0.060863247219&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.391442488392&{}:{}&0.397113876604&{}:{}&0.211443635003&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.114838102404&{}:{}&1.073665974494&{}:{}&-0.188504076898&. \end{alignedat} \]
3b (123)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3b}}&{}\approx{}&-0.135274222954&{}:{}&1.495854294235&{}:{}&-0.360580071281&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&2.680881290508&{}:{}&2.719723059095&{}:{}&-4.400604349603&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.151564971415&{}:{}&1.396802246667&{}:{}&-0.245237275253&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.878107499420&{}:{}&1.878107499420&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.280493836866&{}:{}&1.432743521885&{}:{}&-0.152249685018&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.015865964599&{}:{}&2.221888000710&{}:{}&-1.206022036111&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002050317308&{}:{}&1.466634479794&{}:{}&-0.464584162486&, \end{alignedat} \]
3b (123)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3b}}&{}\approx{}&0.183349467982&{}:{}&1.593331342029&{}:{}&-0.776680810011&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.951056516295&{}:{}&-0.951056516295&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.309016994375&{}:{}&0.000000000000&{}:{}&1.309016994375&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.103197753333&{}:{}&0.896802246667&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3b}}&{}\approx{}&-0.068234650416&{}:{}&1.824968519586&{}:{}&-0.756733869169&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.280493836866&{}:{}&1.432743521885&{}:{}&-0.152249685018&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.015865964599&{}:{}&2.221888000710&{}:{}&-1.206022036111&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002050317308&{}:{}&1.466634479794&{}:{}&-0.464584162486&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&1.951056516295&{}:{}&-0.951056516295&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.309016994375&{}:{}&0.000000000000&{}:{}&1.309016994375&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.103197753333&{}:{}&0.896802246667&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)