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