Problems(1)
Paul Revere is currently at (x0,y0) in the Cartesian plane, which is inside a triangle-shaped ship with vertices at (−257,2524),(−54,53), and (54,−53). Revere has a tea crate in his hands, and there is a second tea crate at (0,0). He must walk to a point on the boundary of the ship to dump the tea, then walk back to pick up the tea crate at the origin. He notices he can take 3 distinct paths to walk the shortest possible distance. Find the ordered pair (x0,y0).Proposed by Derek ZhaoSolution. (−257,256)
Let L, M, and N be the midpoints of BC, AC, and AB, respectively. Let points D, E, and F be the reflections of O=(0,0) over BC, AC, and AB, respectively. Notice since MN∥BC, BC∥EF. Therefore, O is the orthocenter of DEF. Notice that (KMN) is the nine-point circle of ABC because it passes through the midpoints and also the nine-point circle of DEF because it passes through the midpoints of the segments connecting a vertex to the orthocenter. Since O is both the circumcenter of ABC and the orthocenter of DEF and the triangles are 180∘ rotations of each other, Revere is at the orthocenter of ABC. The answer results from adding the vectors OA+OB+OC, which gives the orthocenter of a triangle. 2023themegeoFAlL