Problem list（13 problems）
- Area of this triangle? "Sliding a vertex along a parallel line does not change the area" — this is shearing
- Area of quadrilateral ABCD? Shearing is the tool for "change the shape, keep the area"
- Area of the shaded part? Spot a pair of parallel lines and try sliding a vertex
- Area of triangle ABD? "Shared apex, bases on one line" means the area ratio equals the base ratio
- Area of triangle APD? Wherever P sits on BC, the answer is the same
- Total area of the blue and red triangles? Put P anywhere inside and the two opposite triangles always sum to half
- Area of the red triangle? It works wherever the point is, and whatever shape the rectangle has
- Area of the shaded triangle? One median halves it, three medians sixth it — always, whatever the triangle's shape
- Where does the halving line cross the bottom edge? A rectangle is halved by any line through its centre
- Area of the lawn? Even if the path zigzags, as long as its width is the same the answer does not change
- Area of the path? For any path of constant width — circular, square or wiggly — the area is "length of the centre line × width"
- Area of the inner quadrilateral? However lopsided the original quadrilateral, joining the midpoints always gives a parallelogram, and its area is always half
- Perimeter of this shape? However many steps there are, the answer is the same
Other ideas
- Ratios — answers without fixing a single number
- 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