Theorem

The sum of the medians lies between $\frac{3}{4}$ of the perimeter and the perimeter

Geometry • Triangle • Medians and Centroid

Formula

\frac{3}{4} (a + b + c) < m_{a} + m_{b} + m_{c} < a + b + c

For a triangle with sides

a

b

c

and medians

m_{a}

m_{b}

m_{c}

to those sides,

\frac{3}{4} (a + b + c) < m_{a} + m_{b} + m_{c} < a + b + c

. The upper bound comes from

2 m_{a} < b + c

, obtained by doubling the median into a diagonal of a parallelogram and applying the triangle inequality; the lower bound comes from

\frac{2}{3} (m_{a} + m_{b}) > c

and its two cousins, read off the triangles that the centroid makes with each side.

How to use it

1On a question that asks which inequality between the medians and the sides always holds, test each option against these two bounds rather than against one triangle.
2An option that survives the bounds but is not implied by them is still not 'always correct' — kill it with a concrete triangle, usually the equilateral one, where $m_{a} + m_{b} + m_{c} = \frac{3}{2} (a + b + c) \approx 0.866 (a + b + c)$ .
3Weakening a true bound is safe: $m_{a} + m_{b} + m_{c} < a + b + c$ also gives $2 (m_{a} + m_{b} + m_{c}) < 3 (a + b + c)$ .

A triangle has perimeter

40

cm. Can its three medians total

34

cm? Can they total

28

cm?

Step-by-step

1
$\frac{3}{4} (40) = 30$ and $a + b + c = 40$ , so the median sum must lie strictly between $30$ cm and $40$ cm
Why: Both bounds are read straight off the perimeter.
2
$30 < 34 < 40$ , so $34$ cm is possible; $28 < 30$ , so $28$ cm is not

Answer

34

cm is possible,

28

cm is not

Watch out

Both bounds are strict and both are sharp — a triangle flattening onto a segment approaches the

\frac{3}{4}

floor, and one flattening onto a spike approaches the ceiling — so neither may be written with

\leq

and neither constant can be improved. In particular the medians never sum to exactly half the perimeter.

Builds on

2 : 1

Also known as

sum of the medians, median sum inequality, medians and perimeter, three quarters of the perimeter bound, sum of medians less than perimeter