Derousseau's Generalization of the Malfatti circles

Test Case in ETC

The test case used in “Encyclopedis of Triangle Centers&rdquot; to search for triangle centers.

\(a:b:c=6:9:13\).


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\(\mathbf{7}\) \((222)\)

Malfatti circles

7 (222)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&, \\ P_{\mathbf{7}}&{}\approx{}&0.795862606597&{}:{}&0.200797616737&{}:{}&0.003339776666&, \\ P^-_{\mathbf{7}}&{}\approx{}&-0.839901868399&{}:{}&0.540088659552&{}:{}&1.299813208847&, \\ P^+_{\mathbf{7}}&{}\approx{}&0.442204697637&{}:{}&0.274153507860&{}:{}&0.283641794503&, \\ Q_{\mathbf{7}}&{}\approx{}&0.549877919767&{}:{}&0.349370715505&{}:{}&0.100751364728&, \\ I^\prime_{\mathbf{7}}&{}\approx{}&0.584611232197&{}:{}&0.359644159359&{}:{}&0.055744608444&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{7}}\)
\(P^-_{\mathbf{7}}\)
\(P^+_{\mathbf{7}}\)
\(Q_{\mathbf{7}}\)
\(I^\prime_{\mathbf{7}}\)
7 (222)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7}}&{}\approx{}&0.275392952380&{}:{}&0.296430155845&{}:{}&0.428176891776&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.440419384691&{}:{}&-0.394661384854&{}:{}&0.954242000163&,\\C^\prime_{\mathbf{7}}&{}\approx{}&2.912245269952&{}:{}&4.368367904927&{}:{}&-6.280613174879&, \\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.765128279731&{}:{}&0.231029116630&{}:{}&0.003842603639&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.442954877516&{}:{}&0.555186296159&{}:{}&0.001858826325&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.326506623684&{}:{}&0.082378228781&{}:{}&0.591115147535&, \\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.487157101029&{}:{}&0.436543219524&{}:{}&1.050613881506&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&1.891030156368&{}:{}&2.035485509397&{}:{}&-2.926515665765&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.523081656194&{}:{}&-0.336361283574&{}:{}&0.813279627379&, \\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.776168801416&{}:{}&0.223831198584&,\\B^*_{\mathbf{7}}&{}\approx{}&0.845147817461&{}:{}&0.000000000000&{}:{}&0.154852182539&,\\C^*_{\mathbf{7}}&{}\approx{}&0.611485965281&{}:{}&0.388514034719&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7}}}{B^\prime_{\mathbf{7}}}{C^\prime_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7}}}{B^{\prime\prime}_{\mathbf{7}}}{C^{\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7}}}{B^{\prime\prime\prime}_{\mathbf{7}}}{C^{\prime\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^*_{\mathbf{7}}}{B^*_{\mathbf{7}}}{C^*_{\mathbf{7}}}\)
7 (222)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7}}}}&{}\approx{}&0.922227151517&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7}}}}&{}\approx{}&2.055290461889&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7}}}}&{}\approx{}&13.590477926441&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.214285714286&{}:{}&0.321428571429&{}:{}&0.464285714286&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.275392952380&{}:{}&0.296430155845&{}:{}&0.428176891776&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.440419384691&{}:{}&-0.394661384854&{}:{}&0.954242000163&,\\C^\prime_{\mathbf{7}}&{}\approx{}&2.912245269952&{}:{}&4.368367904927&{}:{}&-6.280613174879&. \end{alignedat} \]
7 (222)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7}}&{}\approx{}&0.795862606597&{}:{}&0.200797616737&{}:{}&0.003339776666&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.765128279731&{}:{}&0.231029116630&{}:{}&0.003842603639&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.442954877516&{}:{}&0.555186296159&{}:{}&0.001858826325&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.326506623684&{}:{}&0.082378228781&{}:{}&0.591115147535&. \end{alignedat} \]
7 (222)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7}}&{}\approx{}&-0.839901868399&{}:{}&0.540088659552&{}:{}&1.299813208847&,\\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.487157101029&{}:{}&0.436543219524&{}:{}&1.050613881506&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&1.891030156368&{}:{}&2.035485509397&{}:{}&-2.926515665765&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.523081656194&{}:{}&-0.336361283574&{}:{}&0.813279627379&. \end{alignedat} \]
7 (222)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7}}&{}\approx{}&0.442204697637&{}:{}&0.274153507860&{}:{}&0.283641794503&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.275392952380&{}:{}&0.296430155845&{}:{}&0.428176891776&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.440419384691&{}:{}&-0.394661384854&{}:{}&0.954242000163&,\\C^\prime_{\mathbf{7}}&{}\approx{}&2.912245269952&{}:{}&4.368367904927&{}:{}&-6.280613174879&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.765128279731&{}:{}&0.231029116630&{}:{}&0.003842603639&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.442954877516&{}:{}&0.555186296159&{}:{}&0.001858826325&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.326506623684&{}:{}&0.082378228781&{}:{}&0.591115147535&, \end{alignedat} \]
7 (222)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7}}&{}\approx{}&0.549877919767&{}:{}&0.349370715505&{}:{}&0.100751364728&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.776168801416&{}:{}&0.223831198584&,\\B^*_{\mathbf{7}}&{}\approx{}&0.845147817461&{}:{}&0.000000000000&{}:{}&0.154852182539&,\\C^*_{\mathbf{7}}&{}\approx{}&0.611485965281&{}:{}&0.388514034719&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7}}&{}\approx{}&0.584611232197&{}:{}&0.359644159359&{}:{}&0.055744608444&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.765128279731&{}:{}&0.231029116630&{}:{}&0.003842603639&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.442954877516&{}:{}&0.555186296159&{}:{}&0.001858826325&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.326506623684&{}:{}&0.082378228781&{}:{}&0.591115147535&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.776168801416&{}:{}&0.223831198584&,\\B^*_{\mathbf{7}}&{}\approx{}&0.845147817461&{}:{}&0.000000000000&{}:{}&0.154852182539&,\\C^*_{\mathbf{7}}&{}\approx{}&0.611485965281&{}:{}&0.388514034719&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Hiroyasu Kamo