Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 120° Top Angle

\(A=30\degree\), \(B=30\degree\), \(C=120\degree\).


[Top] > Isosceles Triangle with 120° Top Angle > 1c (112)

1c(112)

Malfatti circles

1c (112)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&, \\ P_{\mathbf{1c}}&{}\approx{}&-0.553708949516&{}:{}&-0.553708949516&{}:{}&2.107417899032&, \\ P^-_{\mathbf{1c}}&{}\approx{}&-0.000000000000&{}:{}&-0.000000000000&{}:{}&1.000000000000&, \\ P^+_{\mathbf{1c}}&{}\approx{}&-1.300344660511&{}:{}&-1.300344660511&{}:{}&3.600689321023&, \\ Q_{\mathbf{1c}}&{}\approx{}&1.490422535542&{}:{}&1.490422535542&{}:{}&-1.980845071085&, \\ I^\prime_{\mathbf{1c}}&{}\approx{}&-1.762033366018&{}:{}&-1.762033366018&{}:{}&4.524066732036&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{1c}}\)
\(P^-_{\mathbf{1c}}\)
\(P^+_{\mathbf{1c}}\)
\(Q_{\mathbf{1c}}\)
\(I^\prime_{\mathbf{1c}}\)
1c (112)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-1.366025403784&{}:{}&2.366025403784&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&0.000000000000&{}:{}&2.366025403784&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-13.522257448825&{}:{}&-13.522257448825&{}:{}&28.044514897649&, \\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-2.481380475435&{}:{}&-1.240690237717&{}:{}&4.722070713152&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.240690237717&{}:{}&-2.481380475435&{}:{}&4.722070713152&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.683012701892&{}:{}&-0.683012701892&{}:{}&2.366025403784&, \\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.931140182336&{}:{}&-0.000000000000&{}:{}&1.931140182336&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-0.931140182336&{}:{}&1.931140182336&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&0.000000000000&{}:{}&1.000000000000&, \\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-3.039058011260&{}:{}&4.039058011260&,\\B^*_{\mathbf{1c}}&{}\approx{}&-3.039058011260&{}:{}&0.000000000000&{}:{}&4.039058011260&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1c}}}{B^\prime_{\mathbf{1c}}}{C^\prime_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^*_{\mathbf{1c}}}{B^*_{\mathbf{1c}}}{C^*_{\mathbf{1c}}}\)
1c (112)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.366025403784&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.366025403784&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1c}}}}&{}\approx{}&-3.623277963258&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.732050807569&{}:{}&3.732050807569&{}:{}&-6.464101615138&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-1.366025403784&{}:{}&2.366025403784&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&0.000000000000&{}:{}&2.366025403784&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-13.522257448825&{}:{}&-13.522257448825&{}:{}&28.044514897649&. \end{alignedat} \]
1c (112)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1c}}&{}\approx{}&-0.553708949516&{}:{}&-0.553708949516&{}:{}&2.107417899032&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-2.481380475435&{}:{}&-1.240690237717&{}:{}&4.722070713152&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.240690237717&{}:{}&-2.481380475435&{}:{}&4.722070713152&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.683012701892&{}:{}&-0.683012701892&{}:{}&2.366025403784&. \end{alignedat} \]
1c (112)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1c}}&{}\approx{}&-0.000000000000&{}:{}&-0.000000000000&{}:{}&1.000000000000&,\\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.931140182336&{}:{}&-0.000000000000&{}:{}&1.931140182336&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-0.931140182336&{}:{}&1.931140182336&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&0.000000000000&{}:{}&1.000000000000&. \end{alignedat} \]
1c (112)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1c}}&{}\approx{}&-1.300344660511&{}:{}&-1.300344660511&{}:{}&3.600689321023&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-1.366025403784&{}:{}&2.366025403784&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.366025403784&{}:{}&0.000000000000&{}:{}&2.366025403784&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-13.522257448825&{}:{}&-13.522257448825&{}:{}&28.044514897649&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-2.481380475435&{}:{}&-1.240690237717&{}:{}&4.722070713152&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.240690237717&{}:{}&-2.481380475435&{}:{}&4.722070713152&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.683012701892&{}:{}&-0.683012701892&{}:{}&2.366025403784&, \end{alignedat} \]
1c (112)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1c}}&{}\approx{}&1.490422535542&{}:{}&1.490422535542&{}:{}&-1.980845071085&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-3.039058011260&{}:{}&4.039058011260&,\\B^*_{\mathbf{1c}}&{}\approx{}&-3.039058011260&{}:{}&0.000000000000&{}:{}&4.039058011260&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1c}}&{}\approx{}&-1.762033366018&{}:{}&-1.762033366018&{}:{}&4.524066732036&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-2.481380475435&{}:{}&-1.240690237717&{}:{}&4.722070713152&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.240690237717&{}:{}&-2.481380475435&{}:{}&4.722070713152&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.683012701892&{}:{}&-0.683012701892&{}:{}&2.366025403784&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-3.039058011260&{}:{}&4.039058011260&,\\B^*_{\mathbf{1c}}&{}\approx{}&-3.039058011260&{}:{}&0.000000000000&{}:{}&4.039058011260&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)