Using Vectors, Prove That the Line Joining the Midpoints of Two Sides of a Triangle Is Parallel to the Third Side and Exactly Half Its Length

A standard vector algebra proof, not tied to a specific exam paper.

Problem

Using vectors, prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and exactly half its length.

Key insight. Once every vertex of the triangle is written as a position vector, the two midpoints become simple averages — and subtracting one from the other produces exactly half of the third side’s own vector, so both the direction and the length fall out of a single subtraction.

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Approach

Let $\vec A$, $\vec B$, $\vec C$ be the position vectors of the vertices of a triangle, taken from some fixed origin. Write the position vectors of the midpoints of two sides — say $AB$ and $AC$ — using the midpoint formula. The vector from one midpoint to the other is then a direct subtraction of these two expressions; simplifying it reveals it’s exactly half of $\vec{BC}$, which proves both the parallelism and the length ratio at once.

Solution

Step 1 — Set up position vectors

Let $\vec A$, $\vec B$, $\vec C$ be the position vectors of the vertices of $\triangle ABC$, measured from any fixed origin.

Step 2 — Write the midpoints of AB and AC

Let $P$ be the midpoint of $AB$ and $Q$ be the midpoint of $AC$:

$$\vec P = \frac{\vec A+\vec B}2, \qquad \vec Q = \frac{\vec A+\vec C}2$$

Step 3 — Find the vector PQ

$$\vec{PQ} = \vec Q-\vec P = \frac{\vec A+\vec C}2-\frac{\vec A+\vec B}2 = \frac{(\vec A+\vec C)-(\vec A+\vec B)}2 = \frac{\vec C-\vec B}2$$

Step 4 — Recognise this as half of BC

The side $BC$, directed from $B$ to $C$, has vector:

$$\vec{BC} = \vec C-\vec B$$

Comparing with Step 3:

$$\vec{PQ} = \frac12\vec{BC}$$

Step 5 — Interpret the result

Since $\vec{PQ}$ is a positive scalar multiple ($\frac12$) of $\vec{BC}$, the two vectors point in the same direction, so segment $PQ$ is parallel to segment $BC$. And since the scalar is $\frac12$, the length of $PQ$ is exactly half the length of $BC$:

$$PQ \parallel BC, \qquad |PQ| = \frac12|BC|$$

This is the midpoint theorem, proved using vector algebra.

Answer

Common mistakes

• Choosing midpoints of the wrong pair of sides. The theorem pairs the midpoints of two sides with the third side — if $P$ and $Q$ are midpoints of $AB$ and $AC$, the parallel/half relationship is with $BC$, not with $AB$ or $AC$ themselves.
• Subtracting the midpoints in the wrong order. $\vec{PQ}=\vec Q-\vec P$ points from $P$ to $Q$; doing $\vec P-\vec Q$ instead gives $-\frac12\vec{BC}$, which still proves parallelism (a vector and its negative point along the same line) but describes the segment in the opposite direction — worth being consistent about which direction is meant.
• Assuming the origin has to be a specific point, like one of the vertices. The proof works for position vectors measured from any fixed origin — placing the origin at a vertex isn’t wrong, but it’s an unnecessary restriction, since the origin’s own position vector cancels out completely in Step 3.

Practise next

• For the triangle with $A=(1,2)$, $B=(5,4)$, $C=(3,8)$, find the vector joining the midpoints of $AB$ and $AC$, and verify it equals half of $\vec{BC}$.