The Ratio Theorem

A point in a given ratio

Let A and B have position vectors a and b. "P divides AB in the ratio 2 : 1" means AP : PB = 2 : 1: cut AB into three equal parts, and P sits two parts from A and one part from B.

With a = (7, 1) and b = (1, 7), P is the point (3, 5).

Walking there

Go from O to A, then two thirds of the way along AB. AB = b − a, so OP $= a + \left(\frac{2}{3}\right)\left(b - a\right)$.

Collect the a terms and the b terms: OP $= \left(\frac{1}{3}\right)a + \left(\frac{2}{3}\right)b = \frac{{a + 2b}}{3}$. With the numbers, $\frac{{a + 2b}}{3} = \frac{{7 + 2, 1 + 14}}{3} = \frac{{9, 15}}{3} = \left(3, 5\right)$.

Notice which end gets which weight. b gets 2, which is the part from A to P, and a gets 1, the part from P to B. Each end is weighted by the part on the far side of P, so the nearer end gets the larger share.

The ratio theorem

The same walk with AP : PB $= \lambda$ : $\mu$ gives OP $= a + \left(\frac{\lambda}{{\lambda + \mu}}\right)\left(b - a\right) = \frac{{\mu a + \lambda b}}{{\lambda + \mu}}$. This is the ratio theorem.

The two weights, $\frac{\mu}{{\lambda + \mu}}$ and $\frac{\lambda}{{\lambda + \mu}}$, add to 1. That is why the sum is divided by $\lambda + \mu$. With $\lambda = \mu = 1$ the theorem gives $\frac{{a + b}}{2}$, the midpoint.

The centroid

The centroid G of triangle ABC lies on each median, two thirds of the way from the vertex. The median from A goes to the midpoint M of BC, where OM $= \frac{{b + c}}{2}$.

G divides AM in the ratio 2 : 1, so OG $= \frac{{1 \times a + \frac{2 \times \left(b + c\right)}{2}}}{3} = \frac{{a + b + c}}{3}$. The other two medians give the same point, which is why the three medians meet.

The usual slips

Swapping the weights. Writing $\frac{{\lambda a + \mu b}}{{\lambda + \mu}}$ gives the point dividing AB in the ratio $\mu$ : $\lambda$, nearer the other end.

Forgetting to divide by $\lambda + \mu$. $\mu a + \lambda b$ on its own is $\left(\lambda + \mu\right) \times$ OP, a point $\lambda + \mu$ times as far from O as P.

Reading $\lambda$ : $\mu$ as a part against the whole. In 2 : 1, AP is $\frac{2}{3}$ of AB, not $\frac{2}{1}$.