For what value of k do A(-1, 4), B(1, 1) and C(5, k) lie on one straight line?

Compare the squares of the sides to name a triangle: equal sides make it isosceles, and $a^2 + b^2 = c^2$ gives a right angle. Area $= \tfrac{1}{2}\left|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right|$, which is $0$ for collinear points.