2001 IMO Shortlist Problems/G3
Let be a triangle with centroid . Determine, with proof, the position of the point in the plane of such that is a minimum, and express this minimum value in terms of the side lengths of .
We claim that the expression is minimized at , resulting it having a value of ( being the side lengths of ).
We will use vectors, with (meaning that ). Note that by Cauchy-Schwarz, and this bound is clearly reached by . Furthermore, equality is only reached when , , are scalar multiples of , , , respectively. This means that is a scalar multiple of , , and , so . (Note that and are linearly independent, since the centroid is not on .)
Now all that remains is to calculate . To calculate , first let be the midpoint of . Then by Stewart's theorem,
Furthermore, , so By similar reasoning, we can calculate and , so