We first prove that AC, DB are parallel 
1 
AC = DB 
By construction. See Copying a line segment for method and proof 
2 
AD = CB 
By construction. Compass width for AD set from CB 
3 
ACBD is a parallelogram. 
A quadrilateral with congruent opposite sides is a parallelogram.

4 
AC, DB are parallel 
Opposite sides of a parallelogram are parallel. 
We next prove that PE, QF are parallel 
5 
PQ = EF 
Drawn with same compass width 
6 
PQ, EF are parallel 
From (4) 
7 
PQFE is a parallelogram. 
A quadrilateral with one pair of opposite sides parallel and congruent is a parallelogram.

8 
PE, QF are parallel 
Opposite sides of a parallelogram are parallel. 
Prove that triangle AQK is similar to and twice the size of APJ 
9 
∠APJ = ∠AQK 
Corresponding angles. AB is a transversal across the parallels PE, QF 
10 
∠AJP = ∠AKQ 
Corresponding angles. AB is a transversal across the parallels PE, QF 
11 
Triangles AQK, APJ are similar 
AAA. ∠PAJ is common to both, and (9), (10). See
Similar triangles test, angleangleangle. 
12 
Triangles AQK is twice the size of APJ 
AP = PQ. Both drawn with same compass width. 
Prove that AJ = JK 
13 
AK is twice AJ 
(11), (12). AQK is similar to, and twice the size of APJ.
All sides of similar triangles are in the same proportion.
See Properties of similar triangles . 
14 
AJ = JK 
From (13), J must be the midpoint of AK. 
We have proved the first two segments along the given line AB are congruent.
We repeat steps 514 for each successive triangle. For example we show that triangle ARL is similar to and three times APJ, and so
AJ is one third AL. We continue until we have shown that all the segments along AB are congruent.

15 
AJ = JK = KL = LM = MB 
By applying the same steps to triangle AQK, ARL etc. 
16 
AB is divided into n equal parts. 
