Problem list（14 problems）
- Area of the inner triangle? Divide all three sides in the same ratio and the middle triangle is always a fixed fraction of the whole — 21/63 here — regardless of the triangle's shape
- Area of triangle DOA? Find a ratio once, then use it again somewhere else in the same figure
- Area of triangle ADE? Know the ratios on two sides and the triangle between them is just two multiplications
- How long is DC? "Same height, so area ratio = base ratio" works in both directions — lengths to areas, and areas back to lengths
- Area of triangle ABD? "Shared apex, bases on one line" means the area ratio equals the base ratio
- Area of the shaded triangle? A regular hexagon is "six equilateral triangles"
- How tall is the tree? Getting a height you cannot measure from a shadow you can
- How long is the pole's shadow? The sun casts parallel rays; a lamp spreads from a point
- How far is O, where the diagonals cross, from AD? The "hourglass" made by crossing diagonals is always similar, as long as there are parallel sides
- How long is DE? The trick with ratios is to convert to "the whole of AB", not DB
- What is the volume of the bottom part? Double the lengths and areas quadruple, volumes go up eightfold
- How much wider is the big circle than the two small ones? Double the length and the area quadruples; triple it and it goes up ninefold
- Area of the shaded triangle? One median halves it, three medians sixth it — always, whatever the triangle's shape
- How long is MN? "Join the midpoints of two sides and you get half the third side
Other ideas
- Shearing — moving a shape without changing its area
- Add, then subtract — an overlap is a counting problem
- Circles and sectors — multiply by π once, at the very end
- Chasing angles — there are only four tools
- Decide how to count, then count
- Flatten the solid before you think about it
- Write the same thing two different ways
- Fold it, turn it, copy it across
- Back to why the formula works