Derousseau's Generalization of the Malfatti circles

Martin's solution

Problem 4331 (proposed by A. Martin) I. Solution by the Proposer, Mathematical Questions with their Solutions, from the “Educational Times”.

\(a:b:c=231:250:289\).

[Top] > Martin's solution > 2b (121)

2b (121)

Malfatti circles

2b (121)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&, \\ P_{\mathbf{2b}}&{}\approx{}&-0.044670050761&{}:{}&1.160744500846&{}:{}&-0.116074450085&, \\ P^-_{\mathbf{2b}}&{}\approx{}&0.173461538462&{}:{}&0.655128205128&{}:{}&0.171410256410&, \\ P^+_{\mathbf{2b}}&{}\approx{}&-0.467910447761&{}:{}&2.141791044776&{}:{}&-0.673880597015&, \\ Q_{\mathbf{2b}}&{}\approx{}&-3.666666666667&{}:{}&9.333333333333&{}:{}&-4.666666666667&, \\ I^\prime_{\mathbf{2b}}&{}\approx{}&-0.400000000000&{}:{}&2.121212121212&{}:{}&-0.721212121212&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{2b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{2b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{2b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{2b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{2b}}\) Radical center of the Malfatti circles
2b (121)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2b}}&{}\approx{}&1.252777777778&{}:{}&1.620370370370&{}:{}&-1.873148148148&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-1.100000000000&{}:{}&3.476190476190&{}:{}&-1.376190476190&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-1.100000000000&{}:{}&1.190476190476&{}:{}&0.909523809524&, \\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-1.100000000000&{}:{}&2.333333333333&{}:{}&-0.233333333333&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.103529411765&{}:{}&1.372549019608&{}:{}&-0.269019607843&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.103529411765&{}:{}&2.690196078431&{}:{}&-1.586666666667&, \\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.334782608696&{}:{}&1.057971014493&{}:{}&0.276811594203&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.304729729730&{}:{}&0.394144144144&{}:{}&0.301126126126&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.304729729730&{}:{}&1.150900900901&{}:{}&-0.455630630631&, \\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.000000000000&{}:{}&-1.000000000000&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.440000000000&{}:{}&0.000000000000&{}:{}&0.560000000000&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.647058823529&{}:{}&1.647058823529&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2b}}}{B^\prime_{\mathbf{2b}}}{C^\prime_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^*_{\mathbf{2b}}}{B^*_{\mathbf{2b}}}{C^*_{\mathbf{2b}}}\)
2b (121)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.750000000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.285714285714&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.285714285714&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.855555555556&{}:{}&-0.925925925926&{}:{}&1.070370370370&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&1.252777777778&{}:{}&1.620370370370&{}:{}&-1.873148148148&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-1.100000000000&{}:{}&3.476190476190&{}:{}&-1.376190476190&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-1.100000000000&{}:{}&1.190476190476&{}:{}&0.909523809524&. \end{alignedat} \]
2b (121)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2b}}&{}\approx{}&-0.044670050761&{}:{}&1.160744500846&{}:{}&-0.116074450085&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-1.100000000000&{}:{}&2.333333333333&{}:{}&-0.233333333333&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.103529411765&{}:{}&1.372549019608&{}:{}&-0.269019607843&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.103529411765&{}:{}&2.690196078431&{}:{}&-1.586666666667&. \end{alignedat} \]
2b (121)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2b}}&{}\approx{}&0.173461538462&{}:{}&0.655128205128&{}:{}&0.171410256410&,\\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.334782608696&{}:{}&1.057971014493&{}:{}&0.276811594203&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.304729729730&{}:{}&0.394144144144&{}:{}&0.301126126126&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.304729729730&{}:{}&1.150900900901&{}:{}&-0.455630630631&. \end{alignedat} \]
2b (121)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2b}}&{}\approx{}&-0.467910447761&{}:{}&2.141791044776&{}:{}&-0.673880597015&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&1.252777777778&{}:{}&1.620370370370&{}:{}&-1.873148148148&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-1.100000000000&{}:{}&3.476190476190&{}:{}&-1.376190476190&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-1.100000000000&{}:{}&1.190476190476&{}:{}&0.909523809524&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-1.100000000000&{}:{}&2.333333333333&{}:{}&-0.233333333333&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.103529411765&{}:{}&1.372549019608&{}:{}&-0.269019607843&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.103529411765&{}:{}&2.690196078431&{}:{}&-1.586666666667&, \end{alignedat} \]
2b (121)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2b}}&{}\approx{}&-3.666666666667&{}:{}&9.333333333333&{}:{}&-4.666666666667&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.000000000000&{}:{}&-1.000000000000&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.440000000000&{}:{}&0.000000000000&{}:{}&0.560000000000&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.647058823529&{}:{}&1.647058823529&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2b}}&{}\approx{}&-0.400000000000&{}:{}&2.121212121212&{}:{}&-0.721212121212&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-1.100000000000&{}:{}&2.333333333333&{}:{}&-0.233333333333&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.103529411765&{}:{}&1.372549019608&{}:{}&-0.269019607843&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.103529411765&{}:{}&2.690196078431&{}:{}&-1.586666666667&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.000000000000&{}:{}&-1.000000000000&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.440000000000&{}:{}&0.000000000000&{}:{}&0.560000000000&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.647058823529&{}:{}&1.647058823529&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)