Ceva's theorem vector proof

Let ABC be a triangle and suppose that P divides BC in the ratio 1 : s and that Q divides CA in the ratio 1 : t and that R divides AB in the ratio 1 : u. Then AP, BQ and CR are concurrent if and only if stu = 1.

Proof:
We can write the line through the points A and P in a parametric form with parameter alpha:

Similarly, we can write the line through the points B and Q in a parametric form with parameter beta:

Since the expressions are in the triangle coordinate form - 

their coefficients will be equal at their point of intersection, thus we can equate coefficients of A and B, then solve for aplha and beta:

 Similarly we work out coefficients of B and C to get:

 [1]

Now we can avoid doing whole procedure again by cyclically permuting all the quantities involved. Hence we get:

    [2]

Expressions [1], [2] are equal if and only if stu=1.