To divide a segment AB into equal parts you do not need to measure it: draw an auxiliary ray, mark as many equal steps on it as the parts you want, and draw parallels down to AB. With 7 cm in 3 parts, each part is 7/3 cm, a value that falls between the 2.3 cm and 2.4 cm marks of a millimeter ruler and that the construction places without reading the ruler. With these marks you can place points on a graduated number line, which is where they are usually needed.
| You want | Divide AB into | You keep |
|---|---|---|
| Three equal parts | n = 3 | Points P1 and P2 |
| A fraction a/b of AB | n = b | Mark number a |
| Sevenths of a unit | n = 7 | The six interior points |
The first row is solved in the example below and the second in the section “How to mark 2/5 of a segment”. The third is for labeling that number line with sevenths.
Why the parallels work
The auxiliary ray and segment AB are two lines that start at A, and the set of parallels cuts both of them. The intercept theorem, usually credited to Thales, says that in this case the corresponding segments are proportional.
The steps AM1, M1M2 and M2M3 are equal by construction, so their lengths are in the ratio 1 : 1 : 1. The parts AP1, P1P2 and P2B have the same ratio, so they are equal. The result does not depend on the angle of the ray or on the compass opening, only on the marks being equal to one another.
Worked example: dividing a 7 cm segment into 3 parts
AB is a 7 cm segment and you want three equal parts without taking any decimal measurement.
- Draw the ray. It leaves A at an angle between 30° and 60° to AB and is long enough for three marks.
- Mark three equal steps. Keeping the same compass opening, mark M1, M2 and M3 one after another starting at A.
- Join M3 to B. That line is the guide for the parallels.
- Draw two parallels. One passes through M1 and the other through M2, both parallel to M3B. They cut AB at P1 and P2.
- Calculate the length. Each part is 7 ÷ 3 = 7/3 cm.
Result: each part is 7/3 cm, that is, 2 1/3 cm or ≈ 2.33 cm (error under 0.01 cm).
Check: 3 × 7/3 = 7, so the three parts add up to the whole segment.
How to mark 2/5 of a segment
Use the same construction with n = 5, that is, five equal marks and the line joining M5 to B. Then draw only the parallel through M2. Where it cuts AB is point P, two fifths of the way from A, and this holds for any length of AB.
Part PB is the rest, 3/5 of AB. Since 2/5 + 3/5 = 5/5, the two parts make up the whole segment. If you drew the five marks and the parallel through M2 lands closer to B than to A, you probably counted from the wrong end.
Where the construction stops helping
If you only need an approximate value, measuring with a ruler and dividing is faster. The construction also does not tell you how long each part is: it gives its position. For that number, use the fraction of a number tool; to draw the unit with bars or circles, see the guide to drawing fractions; and for scales, the guide to fractions and map scales. Remember that all the parts must come from the same whole.
Sources
Wikipedia states the intercept theorem this way: when two rays from a common point are cut by a pair of parallel lines, the ratio of any two segments on the first ray equals the ratio of the corresponding segments on the second ray. This is what the construction uses (accessed October 17, 2026).