In the Math Corner of my September Diary I included the following brainteaser: "an easy one this month," I added. "You should be able to do this one by just staring hard at the diagram and summoning up some very basic geometry. If you have recourse to pencil and paper, you're overthinking it."
ABCD is a perfect square one unit on a side, P is the midpoint of AB, and PD meets AC at Q. What is the area of the triangle QCD?
Similar triangles QCD, QAP, the former with twice the dimensions of the latter (because CD is twice AP). Drop a perpendicular from Q to CD, another from Q to AP, the former is twice the length of the latter, so the latter is two-thirds of a unit. Area of a triangle is half base times height … Answer: One-third of a square unit.