Find the Point Closest to All Three Corners
Pierre de Fermat challenged Evangelista Torricelli with a deceptively simple question: inside any triangle, find the single point that minimizes the total distance to all three corners added together. It isn't the centroid, and it isn't any of the triangle centers you'd learn in school geometry. Where is it, and how would you even construct it?
Reveal the answer
Build an equilateral triangle outward on two of the original triangle's sides, then draw a line from each new outer vertex to the opposite original corner — the two lines cross exactly at the answer, the Fermat (or Fermat-Torricelli) point, where each side of the triangle subtends a 120-degree angle. Torricelli solved it soon after receiving Fermat's challenge; his pupil Viviani published the proof in 1659.