A triangle has sides 6 cm, 8 cm and 10 cm. What is the sum of the squares of the lengths of its three medians?

Applying Apollonius' theorem to each median and adding gives the standard result: the sum of the squares of the medians is ¾ of the sum of the squares of the sides. That is ¾ × $(36+64+100)$ = ¾ $\times 200=150{\text{cm}}^2$. Checking directly with coordinates (0,0), (8,0), (0,6), the medians are 5, √73 and √52, and 25 + 73 + 52 = 150.