Geometry becomes algebra when you turn points into vectors
No diagrams, no "by inspection", no appeals to symmetry. Position vectors turn classical theorems into algebraic identities that work in any dimension. Drag the slider to morph a quadrilateral and watch the diagonals stay perpendicular — the rhombus condition enforced by the scalar product.
Rhombus Diagonal TestSquare
Drag to adjust side ratio • scalar product stays zero
AC · BD = 0
0.51.01.52.0
Diagonal AC length
1.41
Diagonal BD length
1.41
Given triangle ABC with position vectors a, b, c, the medians meet at the centroid G = (a + b + c)/3. To prove: show that G lies on each median. The midpoint of BC is m = (b + c)/2. The median from A is the line r = a + λ(m − a). Substitute G: we need (a + b + c)/3 = a + λ((b + c)/2 − a). Solving gives λ = 2/3, confirming G lies on this median. The other two medians follow by symmetry. No diagram needed — the algebra forces concurrency.
Know This
Vector proofs replace geometric intuition with algebraic identities: equal sides become equal magnitudes, perpendicular lines become zero scalar products.