51-Many new circle in a triangle

Problem 1:

a- Euler line tangent with the circle through (X(381),X(98),X(597)) at X(381)
b- Euler line tangent with the circle through (X(3),X(98),X(597)) at X(3)
Since a,b we have if I,X(381),J, (X3) are harmonic range then I,J,X(98),X(597) lie on a circle

Problem 2: X(74)= intersection of Jerabek hyperbola with circumcircle. The line though X(74) and parallel with X(13)X(14) meets circumcircle again, meets Euler line, meest Brocard axis at M,Q,P respectively. T is reflection of X(3) on Q. Then we have problem
a- Euler line tangent with the circle through X(3),M,P
b- Euler line tangent with the circle through Q,M,P
Since a,b we have if I,X(3),J,T are harmonic range then I,J,M,P lie on a circle

Problem 3: X(74)= intersection of Jerabek hyperbola with circumcircle. The line though X(74) and parallel with X(13)X(14) meets Euler line, Brocard axis at Q,P respectively. T is reflection of X(3) on P. Then we have problem
a- Brocard axis tangent with the circle through X(74),X3,Q at X(3)
b- Brocard axis tangent with the circle through X(74),Q,T at T
Since a,b we have if I,X(3),J,T are harmonic range then I,J,X(74),Q lie on a circle

Problem 4: The line X(2)X(110) meets Jerabek hyperbolar at D,E then tangent of Jerabek hyperbola at D,E parallel with Brocard axis.
a-The circle through D,E,X(3) tangent with Brocard axis at X(3)
b- Circle through D,E,X(6) tangent with Brocard axis at X(6)
c- D,E, lie on a circke, special D,E,X(15),X(16) lie on a circle.

Problem 5: The line X(2)X(110) meets Jerabek hyperbolar at D,E,X(13),X(14) lie on a circle

Problem 6: Let ABC be a triangle, Let a rectangular circumhyperbola, Center is J. The line is isogonal conjugate of the hyperbolar meets the hyperbolar again at D,E. The hyperbolar meets circle again at M. (d) is a line through M and parallel with EJ. d meets the hyperbolar and the circumcircle at N,P. Then D,E,N,P lie on a circle

Problem 7: 
Xn is reflection of X(1) on X(11) . Then circle through X(1),X(n),X(4) tangent with the line through X(4)X(8) at X(4). Then circle through X(1),X(n),X(8) tangent with the line through X(4)X(8) at X(8)