Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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7b(323)

Malfatti circles

7b (323)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&, \\ P_{\mathbf{7b}}&{}\approx{}&-0.004871864768&{}:{}&1.009698946980&{}:{}&-0.004827082212&, \\ P^-_{\mathbf{7b}}&{}\approx{}&0.012619114489&{}:{}&0.963132629919&{}:{}&0.024248255592&, \\ P^+_{\mathbf{7b}}&{}\approx{}&-0.023212791643&{}:{}&1.058528083615&{}:{}&-0.035315291972&, \\ Q_{\mathbf{7b}}&{}\approx{}&0.134193585987&{}:{}&0.788686619007&{}:{}&0.077119795006&, \\ I^\prime_{\mathbf{7b}}&{}\approx{}&-0.069743418466&{}:{}&1.159367009379&{}:{}&-0.089623590914&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{7b}}\)
\(P^-_{\mathbf{7b}}\)
\(P^+_{\mathbf{7b}}\)
\(Q_{\mathbf{7b}}\)
\(I^\prime_{\mathbf{7b}}\)
7b (323)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7b}}&{}\approx{}&1.848570869931&{}:{}&3.394283479725&{}:{}&-4.242854349656&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.018955349076&{}:{}&1.050547597535&{}:{}&-0.031592248460&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.578480113194&{}:{}&0.771306817591&{}:{}&0.807173295602&, \\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.036707872805&{}:{}&1.041687884990&{}:{}&-0.004980012186&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.129083788468&{}:{}&1.256981028805&{}:{}&-0.127897240337&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.005152544207&{}:{}&1.067870047264&{}:{}&-0.062717503057&, \\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.017915509571&{}:{}&0.992917379863&{}:{}&0.024998129708&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.210184888586&{}:{}&0.385934402960&{}:{}&0.403880708453&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.013328338662&{}:{}&1.017262968701&{}:{}&-0.030591307363&, \\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.910927207563&{}:{}&0.089072792437&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.635045378369&{}:{}&0.000000000000&{}:{}&0.364954621631&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.145407372767&{}:{}&0.854592627233&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7b}}}{B^\prime_{\mathbf{7b}}}{C^\prime_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^*_{\mathbf{7b}}}{B^*_{\mathbf{7b}}}{C^*_{\mathbf{7b}}}\)
7b (323)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7b}}}}&{}\approx{}&-3.394283479725&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.025273798768&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.771306817591&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&1.848570869931&{}:{}&3.394283479725&{}:{}&-4.242854349656&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.018955349076&{}:{}&1.050547597535&{}:{}&-0.031592248460&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.578480113194&{}:{}&0.771306817591&{}:{}&0.807173295602&. \end{alignedat} \]
7b (323)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7b}}&{}\approx{}&-0.004871864768&{}:{}&1.009698946980&{}:{}&-0.004827082212&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.036707872805&{}:{}&1.041687884990&{}:{}&-0.004980012186&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.129083788468&{}:{}&1.256981028805&{}:{}&-0.127897240337&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.005152544207&{}:{}&1.067870047264&{}:{}&-0.062717503057&. \end{alignedat} \]
7b (323)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7b}}&{}\approx{}&0.012619114489&{}:{}&0.963132629919&{}:{}&0.024248255592&,\\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.017915509571&{}:{}&0.992917379863&{}:{}&0.024998129708&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.210184888586&{}:{}&0.385934402960&{}:{}&0.403880708453&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.013328338662&{}:{}&1.017262968701&{}:{}&-0.030591307363&. \end{alignedat} \]
7b (323)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7b}}&{}\approx{}&-0.023212791643&{}:{}&1.058528083615&{}:{}&-0.035315291972&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&1.848570869931&{}:{}&3.394283479725&{}:{}&-4.242854349656&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.018955349076&{}:{}&1.050547597535&{}:{}&-0.031592248460&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.578480113194&{}:{}&0.771306817591&{}:{}&0.807173295602&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.036707872805&{}:{}&1.041687884990&{}:{}&-0.004980012186&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.129083788468&{}:{}&1.256981028805&{}:{}&-0.127897240337&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.005152544207&{}:{}&1.067870047264&{}:{}&-0.062717503057&, \end{alignedat} \]
7b (323)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7b}}&{}\approx{}&0.134193585987&{}:{}&0.788686619007&{}:{}&0.077119795006&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.910927207563&{}:{}&0.089072792437&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.635045378369&{}:{}&0.000000000000&{}:{}&0.364954621631&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.145407372767&{}:{}&0.854592627233&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7b}}&{}\approx{}&-0.069743418466&{}:{}&1.159367009379&{}:{}&-0.089623590914&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.036707872805&{}:{}&1.041687884990&{}:{}&-0.004980012186&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.129083788468&{}:{}&1.256981028805&{}:{}&-0.127897240337&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.005152544207&{}:{}&1.067870047264&{}:{}&-0.062717503057&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.910927207563&{}:{}&0.089072792437&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.635045378369&{}:{}&0.000000000000&{}:{}&0.364954621631&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.145407372767&{}:{}&0.854592627233&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)