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 > 5a (213)

5a(213)

Malfatti circles

5a (213)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&, \\ P_{\mathbf{5a}}&{}\approx{}&1.120652109335&{}:{}&-0.001566642846&{}:{}&-0.119085466488&, \\ P^-_{\mathbf{5a}}&{}\approx{}&0.858564880576&{}:{}&0.091831131859&{}:{}&0.049603987565&, \\ P^+_{\mathbf{5a}}&{}\approx{}&1.495854294235&{}:{}&-0.135274222954&{}:{}&-0.360580071281&, \\ Q_{\mathbf{5a}}&{}\approx{}&1.593331342029&{}:{}&0.183349467982&{}:{}&-0.776680810011&, \\ I^\prime_{\mathbf{5a}}&{}\approx{}&1.824968519586&{}:{}&-0.068234650416&{}:{}&-0.756733869169&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{5a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{5a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{5a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{5a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{5a}}\) Radical center of the Malfatti circles
5a (213)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5a}}&{}\approx{}&1.396802246667&{}:{}&-0.151564971415&{}:{}&-0.245237275253&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&2.719723059095&{}:{}&2.680881290508&{}:{}&-4.400604349603&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.878107499420&{}:{}&-1.878107499420&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.221888000710&{}:{}&-0.015865964599&{}:{}&-1.206022036111&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.432743521885&{}:{}&-0.280493836866&{}:{}&-0.152249685018&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.466634479794&{}:{}&-0.002050317308&{}:{}&-0.464584162486&, \\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.397113876604&{}:{}&0.391442488392&{}:{}&0.211443635003&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.053444473822&{}:{}&-0.114307721041&{}:{}&0.060863247219&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.073665974494&{}:{}&0.114838102404&{}:{}&-0.188504076898&, \\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.309016994375&{}:{}&1.309016994375&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.951056516295&{}:{}&0.000000000000&{}:{}&-0.951056516295&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.896802246667&{}:{}&0.103197753333&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5a}}}{B^\prime_{\mathbf{5a}}}{C^\prime_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^*_{\mathbf{5a}}}{B^*_{\mathbf{5a}}}{C^*_{\mathbf{5a}}}\)
5a (213)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5a}}}}&{}\approx{}&-0.245237275253&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5a}}}}&{}\approx{}&-4.400604349603&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5a}}}}&{}\approx{}&-3.038841768588&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.618033988750&{}:{}&0.618033988750&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.396802246667&{}:{}&-0.151564971415&{}:{}&-0.245237275253&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&2.719723059095&{}:{}&2.680881290508&{}:{}&-4.400604349603&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.878107499420&{}:{}&-1.878107499420&{}:{}&1.000000000000&. \end{alignedat} \]
5a (213)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5a}}&{}\approx{}&1.120652109335&{}:{}&-0.001566642846&{}:{}&-0.119085466488&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.221888000710&{}:{}&-0.015865964599&{}:{}&-1.206022036111&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.432743521885&{}:{}&-0.280493836866&{}:{}&-0.152249685018&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.466634479794&{}:{}&-0.002050317308&{}:{}&-0.464584162486&. \end{alignedat} \]
5a (213)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5a}}&{}\approx{}&0.858564880576&{}:{}&0.091831131859&{}:{}&0.049603987565&,\\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.397113876604&{}:{}&0.391442488392&{}:{}&0.211443635003&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.053444473822&{}:{}&-0.114307721041&{}:{}&0.060863247219&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.073665974494&{}:{}&0.114838102404&{}:{}&-0.188504076898&. \end{alignedat} \]
5a (213)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5a}}&{}\approx{}&1.495854294235&{}:{}&-0.135274222954&{}:{}&-0.360580071281&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.396802246667&{}:{}&-0.151564971415&{}:{}&-0.245237275253&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&2.719723059095&{}:{}&2.680881290508&{}:{}&-4.400604349603&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.878107499420&{}:{}&-1.878107499420&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.221888000710&{}:{}&-0.015865964599&{}:{}&-1.206022036111&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.432743521885&{}:{}&-0.280493836866&{}:{}&-0.152249685018&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.466634479794&{}:{}&-0.002050317308&{}:{}&-0.464584162486&, \end{alignedat} \]
5a (213)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5a}}&{}\approx{}&1.593331342029&{}:{}&0.183349467982&{}:{}&-0.776680810011&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.309016994375&{}:{}&1.309016994375&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.951056516295&{}:{}&0.000000000000&{}:{}&-0.951056516295&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.896802246667&{}:{}&0.103197753333&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5a}}&{}\approx{}&1.824968519586&{}:{}&-0.068234650416&{}:{}&-0.756733869169&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&2.221888000710&{}:{}&-0.015865964599&{}:{}&-1.206022036111&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.432743521885&{}:{}&-0.280493836866&{}:{}&-0.152249685018&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.466634479794&{}:{}&-0.002050317308&{}:{}&-0.464584162486&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.309016994375&{}:{}&1.309016994375&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.951056516295&{}:{}&0.000000000000&{}:{}&-0.951056516295&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.896802246667&{}:{}&0.103197753333&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)