Derousseau's Generalization of the Malfatti circles

Half of an Equilateral Triangle

\(A=30\degree\), \(B=60\degree\), \(C=90\degree\).


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[Lob & Richmond]

\(\mathbf{5}\) \((202)\)

Malfatti circles

5 (202)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&, \\ P_{\mathbf{5}}&{}\approx{}&0.973097836996&{}:{}&0.000561995317&{}:{}&0.026340167687&, \\ P^-_{\mathbf{5}}&{}\approx{}&1.100769080968&{}:{}&-0.060688757499&{}:{}&-0.040080323469&, \\ P^+_{\mathbf{5}}&{}\approx{}&0.877477913028&{}:{}&0.046436008487&{}:{}&0.076086078486&, \\ Q_{\mathbf{5}}&{}\approx{}&0.779689971461&{}:{}&-0.027504484097&{}:{}&0.247814512636&, \\ I^\prime_{\mathbf{5}}&{}\approx{}&0.790951032877&{}:{}&0.025016007943&{}:{}&0.184032959181&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{5}}\)
\(P^-_{\mathbf{5}}\)
\(P^+_{\mathbf{5}}\)
\(Q_{\mathbf{5}}\)
\(I^\prime_{\mathbf{5}}\)
5 (202)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5}}&{}\approx{}&0.892793641318&{}:{}&0.049754644217&{}:{}&0.057451714465&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.777317076691&{}:{}&-1.331951230074&{}:{}&1.554634153382&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.624844448887&{}:{}&1.082262332300&{}:{}&-0.707106781187&, \\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.692790739919&{}:{}&0.006417705730&{}:{}&0.300791554352&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.881016683009&{}:{}&0.095135635614&{}:{}&0.023847681377&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.888678273303&{}:{}&0.000513240303&{}:{}&0.110808486394&, \\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-0.541953664682&{}:{}&0.928650446468&{}:{}&0.613303218214&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.981048146442&{}:{}&0.054672994102&{}:{}&-0.035721140544&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.990866739077&{}:{}&-0.054629506117&{}:{}&0.063762767040&, \\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.124844448887&{}:{}&1.124844448887&,\\B^*_{\mathbf{5}}&{}\approx{}&0.758819045103&{}:{}&0.000000000000&{}:{}&0.241180954897&,\\C^*_{\mathbf{5}}&{}\approx{}&1.036566092485&{}:{}&-0.036566092485&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5}}}{B^\prime_{\mathbf{5}}}{C^\prime_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5}}}{B^{\prime\prime}_{\mathbf{5}}}{C^{\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5}}}{B^{\prime\prime\prime}_{\mathbf{5}}}{C^{\prime\prime\prime}_{\mathbf{5}}}\)
\(\triangle{A^*_{\mathbf{5}}}{B^*_{\mathbf{5}}}{C^*_{\mathbf{5}}}\)
5 (202)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5}}}}&{}\approx{}&0.135932215914&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5}}}}&{}\approx{}&3.678303900494&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5}}}}&{}\approx{}&2.956795678960&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.211324865405&{}:{}&0.366025403784&{}:{}&0.422649730810&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.892793641318&{}:{}&0.049754644217&{}:{}&0.057451714465&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.777317076691&{}:{}&-1.331951230074&{}:{}&1.554634153382&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.624844448887&{}:{}&1.082262332300&{}:{}&-0.707106781187&. \end{alignedat} \]
5 (202)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5}}&{}\approx{}&0.973097836996&{}:{}&0.000561995317&{}:{}&0.026340167687&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.692790739919&{}:{}&0.006417705730&{}:{}&0.300791554352&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.881016683009&{}:{}&0.095135635614&{}:{}&0.023847681377&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.888678273303&{}:{}&0.000513240303&{}:{}&0.110808486394&. \end{alignedat} \]
5 (202)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5}}&{}\approx{}&1.100769080968&{}:{}&-0.060688757499&{}:{}&-0.040080323469&,\\ A^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&-0.541953664682&{}:{}&0.928650446468&{}:{}&0.613303218214&,\\B^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.981048146442&{}:{}&0.054672994102&{}:{}&-0.035721140544&,\\C^{\prime\prime\prime}_{\mathbf{5}}&{}\approx{}&0.990866739077&{}:{}&-0.054629506117&{}:{}&0.063762767040&. \end{alignedat} \]
5 (202)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5}}&{}\approx{}&0.877477913028&{}:{}&0.046436008487&{}:{}&0.076086078486&,\\ A^\prime_{\mathbf{5}}&{}\approx{}&0.892793641318&{}:{}&0.049754644217&{}:{}&0.057451714465&,\\B^\prime_{\mathbf{5}}&{}\approx{}&0.777317076691&{}:{}&-1.331951230074&{}:{}&1.554634153382&,\\C^\prime_{\mathbf{5}}&{}\approx{}&0.624844448887&{}:{}&1.082262332300&{}:{}&-0.707106781187&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.692790739919&{}:{}&0.006417705730&{}:{}&0.300791554352&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.881016683009&{}:{}&0.095135635614&{}:{}&0.023847681377&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.888678273303&{}:{}&0.000513240303&{}:{}&0.110808486394&, \end{alignedat} \]
5 (202)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5}}&{}\approx{}&0.779689971461&{}:{}&-0.027504484097&{}:{}&0.247814512636&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.124844448887&{}:{}&1.124844448887&,\\B^*_{\mathbf{5}}&{}\approx{}&0.758819045103&{}:{}&0.000000000000&{}:{}&0.241180954897&,\\C^*_{\mathbf{5}}&{}\approx{}&1.036566092485&{}:{}&-0.036566092485&{}:{}&0.000000000000&. \end{alignedat} \]
5 (202)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5}}&{}\approx{}&0.790951032877&{}:{}&0.025016007943&{}:{}&0.184032959181&,\\ A^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.692790739919&{}:{}&0.006417705730&{}:{}&0.300791554352&,\\B^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.881016683009&{}:{}&0.095135635614&{}:{}&0.023847681377&,\\C^{\prime\prime}_{\mathbf{5}}&{}\approx{}&0.888678273303&{}:{}&0.000513240303&{}:{}&0.110808486394&,\\ A^*_{\mathbf{5}}&{}\approx{}&0.000000000000&{}:{}&-0.124844448887&{}:{}&1.124844448887&,\\B^*_{\mathbf{5}}&{}\approx{}&0.758819045103&{}:{}&0.000000000000&{}:{}&0.241180954897&,\\C^*_{\mathbf{5}}&{}\approx{}&1.036566092485&{}:{}&-0.036566092485&{}:{}&0.000000000000&. \end{alignedat} \]
5 (202)

Hiroyasu Kamo