The intercept theorem and midsegments: transferring a division

You need five equal parts along a segment, but equal steps are easier to mark in another direction. Parallel lines let you transfer those marks while preserving their ratios.

Parallel lines cut proportional corresponding lengths on two transversals. A triangle’s midsegment is half its third side; a trapezoid’s midsegment is the average of its bases. You will use parallelism, quadrilaterals and ordinary numerical ratios.

One name, different statements

Ukrainian school texts often call the equal-intercept statement Thales’ theorem: if parallel lines cut equal consecutive segments on one transversal, they cut equal segments on another. In English, Thales’ theorem often means a different fact: an angle inscribed over a diameter is a right angle. We covered it in G008. AMSI.

The exact conditions matter more than the name: which lines are parallel, and which segments correspond?

From equal parts to proportions

Take two rays from O with an angle strictly between 0° and 180°. Two distinct parallel lines meet the first ray in order O,A,B and the second in order O,A′,B′. Then:

$\frac{OA}{AB}=\frac{OA'}{A'B'},\qquad \frac{OA}{OB}=\frac{OA'}{OB'}.$

For example, OA=3, AB=2 and OA′=4.5 in the same units. From 3/2=4.5/A′B′, we obtain A′B′=3. The full lengths OB=5 and OB′=7.5 also have the corresponding ratio 5/7.5=3/4.5. The scale change on the other ray is constant throughout. Do not mix a part with a whole or reverse the order in just one fraction.

Corresponding parts: 3 : 2 = 4.5 : 3

Triangle: MN = AC / 2

Trapezoid: 3 + 6 = 9

Five equal steps → five parts

Uniform scale within each panel, arbitrary units. First: O,A,B and O,A′,B′ in corresponding order, AA′∥BB′; numbers give lengths along the rays. Second: M,N are midpoints of AB,BC. Third: M,N are midpoints of AD,BC; K is the midpoint of AC, MK=3, KN=6. Fourth: five equal auxiliary steps and five equal parts of AB; the sloping connecting segments are parallel to X₅B.

This is the intercept theorem, with equal steps as a special case. It also holds for three or more distinct parallel lines and two transversals: corresponding consecutive intercepts have equal ratios. Our numbers only check an application. The general theorem does not require a common unit that fits into every segment a whole number of times; Euclid VI.2 proves proportional division using area ratios.

A triangle’s midsegment: half a side

In triangle ABC, let M be the midpoint of AB and N the midpoint of BC. Segment MN is a midsegment. Then MN∥AC and MN=AC/2. If AC=10, the midsegment is 5.

We can prove this with familiar tools. Extend MN beyond N to E with NE=MN. Triangles MBN and ECN are congruent by two sides and the vertical angle between them: BN=NC and MN=NE. Hence MB=EC and MB∥EC. Since AM=MB, quadrilateral AMEC is a parallelogram: AM and EC are equal and parallel. Thus ME∥AC, ME=AC, and MN is half of ME.

The converse is useful too: a line through the midpoint of one side, parallel to another side, passes through the midpoint of the third. A triangle has three midsegments, depending on the chosen pair of sides. AMSI on midsegments.

In a trapezoid, average the two bases

Let AB∥CD, with M and N the midpoints of the legs AD and BC. Then:

$MN\parallel AB,\qquad MN=\frac{AB+CD}{2}.$

Bases 12 and 6 give MN=9. Why not 6? Draw diagonal AC and mark its midpoint K. In triangle ADC, midsegment MK is CD/2=3. In triangle ABC, midsegment KN is AB/2=6. Both are parallel to the bases and lie on one line, giving MN=3+6=9. Joining the midpoints of the bases themselves produces a different segment; this formula does not apply to it.

Five equal parts without dividing on a ruler

For a given AB:

Draw any auxiliary ray from A that does not lie on line AB.
With a compass, mark five equal nonzero steps to X₁,…,X₅. The step length is arbitrary, but keep the compass opening fixed.
Join X₅ to B.
Through X₁,…,X₄ draw lines parallel to X₅B. Their intersections with AB divide it into five equal parts.

The equal-intercept theorem justifies this. If AB=17 cm, each part is 17/5=3.4 cm, but the construction does not require measuring 3.4 cm first. Construct the parallels by copying an angle as in G010. The ideal construction is exact; tool errors accumulate in a physical drawing.