Vectors & Matrices

Any point on the line through $A$ and $B$ is $\mathbf{r}=(1-\alpha )\mathbf{a}+\alpha \mathbf{b}$, and the single condition that the weights sum to $1$ is exactly the test for collinearity. Read geometry straight off the weights: $\alpha$ is the fraction of the way from $A$ to $B$, and a point dividing $AB$ in ratio $p:q$ is $\frac{q\mathbf{a}+p\mathbf{b}}{p+q}$.

Trigger: a point is described by a ratio on a segment, lies on a join of two named points, or you must prove three points are collinear.

Instances:
(i) Place a point dividing $AB$ in a given ratio, then impose a further condition (angle bisector, equal lengths) to pin the parameter.
(ii) Prove collinearity or find where a constructed line cuts a side by forcing the three weights of a point to sum to $1$.
(iii) Write a point as a weighted average of three vertices, $\mathbf{r}=\alpha \mathbf{a}+\beta \mathbf{b}+\gamma \mathbf{c}$ with $\alpha +\beta +\gamma =1$, to land it in a plane or on a cevian, and recover side-ratios from the weights.