Y by baopbc, mineiraojose, enhanced, amar_04, Paramizo_Dicrominique, Adventure10, Mango247
Given a
and a point
lying on the Neuberg cubic of
Let
be the 2nd / 1st Fermat point of
and
be the 1st / 2nd Fermat point of
respectively. Prove that
(a)
are concyclic.
(b)
are concurrent.
![$ \triangle ABC $](http://latex.artofproblemsolving.com/c/d/8/cd8e45b7154d5c41b30124b766edccc03c41ddad.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ \triangle ABC. $](http://latex.artofproblemsolving.com/2/8/9/2894ae467930d3efcad8db63b802ea1dae56c2f6.png)
![$ S $](http://latex.artofproblemsolving.com/a/9/7/a9759b74d386d1ef46b532c5d93e14392ff27100.png)
![$ \triangle ABC $](http://latex.artofproblemsolving.com/c/d/8/cd8e45b7154d5c41b30124b766edccc03c41ddad.png)
![$ T, $](http://latex.artofproblemsolving.com/8/9/3/893f1c9ff14ee0f6e3cb6d290a00d747940fafd1.png)
![$ T_a, $](http://latex.artofproblemsolving.com/6/2/7/6273cbbf2a107da7b821850f33f149394a5568d7.png)
![$ T_b, $](http://latex.artofproblemsolving.com/7/9/2/792e4080f5e998921b77ff6526e8a748f3bb2082.png)
![$ T_c $](http://latex.artofproblemsolving.com/e/b/b/ebb25435624ad721cecf569c429ee6e231b770ea.png)
![$ \triangle ABC, $](http://latex.artofproblemsolving.com/f/a/d/fadb8414d426ea65e63cf49165c2b1d0e255abce.png)
![$ \triangle BPC, $](http://latex.artofproblemsolving.com/9/3/c/93c979fa518516d1d57c6b9eace5d928826f3eff.png)
![$ \triangle CPA, $](http://latex.artofproblemsolving.com/3/5/c/35c0a203a290c464c1848406fa7102ea9d19fe19.png)
![$ \triangle APB, $](http://latex.artofproblemsolving.com/3/7/f/37fd5d2ecc089a40574bcd1a30da1a04c8fd0419.png)
(a)
![$ T, $](http://latex.artofproblemsolving.com/8/9/3/893f1c9ff14ee0f6e3cb6d290a00d747940fafd1.png)
![$ T_a, $](http://latex.artofproblemsolving.com/6/2/7/6273cbbf2a107da7b821850f33f149394a5568d7.png)
![$ T_b, $](http://latex.artofproblemsolving.com/7/9/2/792e4080f5e998921b77ff6526e8a748f3bb2082.png)
![$ T_c $](http://latex.artofproblemsolving.com/e/b/b/ebb25435624ad721cecf569c429ee6e231b770ea.png)
(b)
![$ AT_a, $](http://latex.artofproblemsolving.com/e/5/5/e55d553ced0b2e669cd81822074c6f4ef074801e.png)
![$ BT_b, $](http://latex.artofproblemsolving.com/6/d/5/6d5b5e96c01cc46cf0afe1460359f84b7e3f2464.png)
![$ CT_c, $](http://latex.artofproblemsolving.com/d/f/f/dffb3f6486da51e10597961262ee60c99b293590.png)
![$ PS $](http://latex.artofproblemsolving.com/8/5/9/8593d3190cd355cabd995d3dae410a0c4cf61b4f.png)