Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


[Top] > The Smallest Pythagorean Triangle > 0c (110)

0c(110)

Malfatti circles

0c (110)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&, \\ P_{\mathbf{c}}&{}\approx{}&0.899207291927&{}:{}&0.991734417275&{}:{}&-0.890941709201&, \\ P^-_{\mathbf{c}}&{}\approx{}&0.785034888947&{}:{}&0.800127390635&{}:{}&-0.585162279583&, \\ P^+_{\mathbf{c}}&{}\approx{}&0.981936569052&{}:{}&1.130572791456&{}:{}&-1.112509360508&, \\ Q_{\mathbf{c}}&{}\approx{}&0.779553414178&{}:{}&0.668448620013&{}:{}&-0.448002034191&, \\ I^\prime_{\mathbf{c}}&{}\approx{}&1.078042283606&{}:{}&1.307293031736&{}:{}&-1.385335315343&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{c}}\)
\(P^-_{\mathbf{c}}\)
\(P^+_{\mathbf{c}}\)
\(Q_{\mathbf{c}}\)
\(I^\prime_{\mathbf{c}}\)
0c (110)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{c}}&{}\approx{}&1.137704095259&{}:{}&0.550816381038&{}:{}&-0.688520476297&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.467232496600&{}:{}&1.311488331067&{}:{}&-0.778720827667&,\\C^\prime_{\mathbf{c}}&{}\approx{}&3.564755078384&{}:{}&4.753006771179&{}:{}&-7.317761849563&, \\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.826177851031&{}:{}&1.710296418385&{}:{}&-1.536474269416&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.389759904956&{}:{}&0.987225219851&{}:{}&-1.376985124807&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.823077303330&{}:{}&0.907770763337&{}:{}&-0.730848066667&, \\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.570086553371&{}:{}&1.600192340823&{}:{}&-1.170278894194&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.353636460621&{}:{}&0.655359455580&{}:{}&-1.008995916200&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.707193012522&{}:{}&0.720788983715&{}:{}&-0.427981996237&, \\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&3.032247551123&{}:{}&-2.032247551123&,\\B^*_{\mathbf{c}}&{}\approx{}&2.351229586822&{}:{}&0.000000000000&{}:{}&-1.351229586822&,\\C^*_{\mathbf{c}}&{}\approx{}&0.538364861216&{}:{}&0.461635138784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{c}}}{B^\prime_{\mathbf{c}}}{C^\prime_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{c}}}{B^{\prime\prime}_{\mathbf{c}}}{C^{\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{c}}}{B^{\prime\prime\prime}_{\mathbf{c}}}{C^{\prime\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^*_{\mathbf{c}}}{B^*_{\mathbf{c}}}{C^*_{\mathbf{c}}}\)
0c (110)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{c}}}}&{}\approx{}&0.275408190519&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{c}}}}&{}\approx{}&0.311488331067&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{c}}}}&{}\approx{}&2.376503385590&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.137704095259&{}:{}&0.550816381038&{}:{}&-0.688520476297&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.467232496600&{}:{}&1.311488331067&{}:{}&-0.778720827667&,\\C^\prime_{\mathbf{c}}&{}\approx{}&3.564755078384&{}:{}&4.753006771179&{}:{}&-7.317761849563&. \end{alignedat} \]
0c (110)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{c}}&{}\approx{}&0.899207291927&{}:{}&0.991734417275&{}:{}&-0.890941709201&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.826177851031&{}:{}&1.710296418385&{}:{}&-1.536474269416&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.389759904956&{}:{}&0.987225219851&{}:{}&-1.376985124807&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.823077303330&{}:{}&0.907770763337&{}:{}&-0.730848066667&. \end{alignedat} \]
0c (110)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{c}}&{}\approx{}&0.785034888947&{}:{}&0.800127390635&{}:{}&-0.585162279583&,\\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.570086553371&{}:{}&1.600192340823&{}:{}&-1.170278894194&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.353636460621&{}:{}&0.655359455580&{}:{}&-1.008995916200&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.707193012522&{}:{}&0.720788983715&{}:{}&-0.427981996237&. \end{alignedat} \]
0c (110)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{c}}&{}\approx{}&0.981936569052&{}:{}&1.130572791456&{}:{}&-1.112509360508&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.137704095259&{}:{}&0.550816381038&{}:{}&-0.688520476297&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.467232496600&{}:{}&1.311488331067&{}:{}&-0.778720827667&,\\C^\prime_{\mathbf{c}}&{}\approx{}&3.564755078384&{}:{}&4.753006771179&{}:{}&-7.317761849563&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.826177851031&{}:{}&1.710296418385&{}:{}&-1.536474269416&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.389759904956&{}:{}&0.987225219851&{}:{}&-1.376985124807&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.823077303330&{}:{}&0.907770763337&{}:{}&-0.730848066667&, \end{alignedat} \]
0c (110)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{c}}&{}\approx{}&0.779553414178&{}:{}&0.668448620013&{}:{}&-0.448002034191&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&3.032247551123&{}:{}&-2.032247551123&,\\B^*_{\mathbf{c}}&{}\approx{}&2.351229586822&{}:{}&0.000000000000&{}:{}&-1.351229586822&,\\C^*_{\mathbf{c}}&{}\approx{}&0.538364861216&{}:{}&0.461635138784&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{c}}&{}\approx{}&1.078042283606&{}:{}&1.307293031736&{}:{}&-1.385335315343&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.826177851031&{}:{}&1.710296418385&{}:{}&-1.536474269416&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.389759904956&{}:{}&0.987225219851&{}:{}&-1.376985124807&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.823077303330&{}:{}&0.907770763337&{}:{}&-0.730848066667&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&3.032247551123&{}:{}&-2.032247551123&,\\B^*_{\mathbf{c}}&{}\approx{}&2.351229586822&{}:{}&0.000000000000&{}:{}&-1.351229586822&,\\C^*_{\mathbf{c}}&{}\approx{}&0.538364861216&{}:{}&0.461635138784&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)