How Do You Use Vectors to Prove Geometric Properties?

You prove geometric properties by representing points with position vectors, translating the required property into a vector relationship, and then establishing that relationship algebraically. This is Mathematics AA HL content from AHL 3.12: Vector definitions, applicable across Papers 1, 2 and 3.

The vector method

Let the position vectors of points $A$ and $B$ be $\mathbf{a}$ and $\mathbf{b}$. Their displacement vector is

$$\overrightarrow{AB}=\mathbf{b}-\mathbf{a}.$$

Once displacement vectors have been formed, use the condition corresponding to the property being proved.

Worked proof: diagonals of a parallelogram

Suppose $ABCD$ is a parallelogram with position vectors $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}$. Since opposite sides are equal as vectors,

$$\overrightarrow{AB}=\overrightarrow{DC}.$$

Therefore,

$$\mathbf{b}-\mathbf{a}=\mathbf{c}-\mathbf{d},$$

so

$$\mathbf{c}=\mathbf{b}+\mathbf{d}-\mathbf{a}.$$

The midpoint of diagonal $AC$ has position vector

$$\frac{\mathbf{a}+\mathbf{c}}{2} =\frac{\mathbf{b}+\mathbf{d}}{2}.$$

The midpoint of diagonal $BD$ also has position vector $\frac{\mathbf{b}+\mathbf{d}}{2}$. Hence the diagonals have the same midpoint, so they bisect each other.

Short example: proving collinearity

Let $A$, $B$, and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$. If

$$\mathbf{b}-\mathbf{a}=2(\mathbf{c}-\mathbf{a}),$$

then $\overrightarrow{AB}=2\overrightarrow{AC}$. Therefore, both displacement vectors have the same direction and share the point , so , , and are collinear. The scalar multiple is the crucial evidence; merely obtaining three related coordinates is not a proof.

This method also distinguishes collinearity from equality of vectors. A negative scalar still proves collinearity but shows that the vectors point in opposite directions.

A common misconception is that a diagram demonstrates the result. It does not: a diagram suggests a property, while a vector equality proves it.

Exam technique

For the command term prove, show a complete logical sequence: define vectors, form the relevant displacement vectors, establish the required vector condition, and finish with an explicit geometric conclusion. Do not stop at an unexplained algebraic equality.