The altitudes of a triangle are concurrent.
Let , and be the vertices of the triangle . If is the slope of , then using two point formula to find slope of line
Since altitude is perpendicular to the side , so its slope is given as using condition of perpendicular slope is
Equation of altitude passing through with slope is
For the equation of altitude , we just replace by , by and by in (iii) (i.e. ), so
For the equation of altitude , we just replace by , by and by in (iv) (i.e. ), so
To see whether the altitudes (iii), (iv) and (v) are concurrent, consider the determinant.
This shows that the altitudes of the triangle are concurrent.