Planimetry: triangles, quadrilaterals, circles, similarity, areas, coordinates and vectors
Triangles, quadrilaterals, circles
Triangle toolkit: angle sum π; an exterior angle equals the sum of the two remote interior angles; the sine rule a/sin A = b/sin B = 2R; the cosine rule a² = b² + c² − 2bc cos A; area = ½ab sin C = ½ah = √(p(p − a)(p − b)(p − c)). Special lines: medians meet at the centroid, which divides each in ratio 2:1; the incenter is equidistant from the sides, the circumcenter from the vertices.
Quadrilaterals: parallelogram area bh; trapezoid ½(a + b)h; rhombus ½d₁d₂; any quadrilateral with perpendicular diagonals has area ½d₁d₂. Circles: an inscribed angle is half the central angle on the same arc; a tangent is perpendicular to the radius; the tangent–chord angle equals the inscribed angle on the other side; intersecting chords satisfy PA·PB = PC·PD.
Similarity: corresponding lengths scale by k, areas by k², volumes by k³. This one fact solves most "area of the similar figure" questions.
Coordinate and vector methods: distance, midpoint, section formula, dot product a·b = |a||b|cos θ, perpendicularity a·b = 0.
Traps. Using the wrong altitude in ½bh. Forgetting the ambiguous SSA case. Assuming a triangle is isosceles or right because the diagram suggests it.
Worked example. Two similar triangles have sides in ratio 2:3, so their areas are in ratio 4:9; if the smaller area is 20, the larger is 45.