If in a triangle A=(1,10),circumcenter=(-1/3,2/3) and orthocentre =(11/3,4/3) then the co-ordinates of the side opposite to A is:

In a triangle, the circumcenter orthocentre and the centroid are collinear and they lie on a line called as Euler line.
The centroid divides the orthocentre and circumcentre in 2:1. Hence, find the coordinates of centroid from this which is ( 1 , 8/9)

Say B ( a,b) and C (c,d)
From centroid :
(1 + a + c)/3 = 1
(10+b + d)/3 = 8/9
From circumcentre :
Distance between circumcentre and A = Distance between circumcentre and B = Distance between circumcentre and C = Radii of circumcircle
Therefore, the other two equations of a,b,c and d can be found from here. Hence,
There are 4 equations and 4 unknowns. I encourage you do these calculations by yourself.

Thanks
Bharat Bajaj
AskIITians Faculty
IIT Delhi