x=[x2(x1−x3)+y2(y1−y3)][y3−y2]−[y3−y1][x1(x2−x3)+y1(y2−y3)] / [(x3−x2)(y3−y1)−(y3−y2)(x3−x1)]
y=[x2(x1−x3)+y2(y1−y3)][x3−x2]−[x3−x1][x1(x2−x3)+y1(y2−y3)] / [(y3−y2)(x3−x1)−(x3−x2)(y3−y1)]
where the vertices of triangle are (x1,y1), (x2,y2), (x3,y3).
The method to find the othrocentre is as follows:
i) It is the point of concurrency of the 3 altitudes of the triangle.
ii) First find slope of any two sides; say AB & AC using slope formula. Then slope of the perpendicular to AB through C will be negative reciprocal of the slope of AB. Then using slope-point form, find the equation of the altitude through C.
iii) Similarly, equation to the altitude through B for the side AC can be found out.
iv) Then solve these two altitude equations, which would give the orthocentre of the triangle.