Question

If the centroid and circumcenter of a triangle are (3,3),(6,2),then the orthocenter is?

Answer 1

Let the Orthocenter be  H(x,y).  Let Centroid be G(3,3).  Let the circumcenter be O(6,2).

Then we have   2 * OG = GH       or  GH = 2 * GO    (to remember better)
GH^2 = 4 GO^2 
so (x - 3)^2 + (y-3)^2 = 4 [ 3^2 + 1^2] = 40
x^2 + y^2 - 6x - 6 y - 22 = 0      --- (1)

The circumcenter O, centroid G and orthocenter H are always collinear.  So  find the slope of the Euler line :
        (y-3)/(x-3) = (2-3)/(6-3) = -1/3
         3 y - 9 =  3 - x
         x = 3 (4 - y)       --- (2)

Substituting for y in (1) we get: 
y^2 - 6 y + 9 (4 - y)^2 - 18 (4 - y) - 22 = 0
y^2 - 6 y + 5 = 0
=>  y = 5 or 1       =>  so   x = -3  or 9
So Orthocenter can be :  (-3,  5)  or  (9, 1)

There are two triangles possible.