On Friday you used an unmarked ruler and a collapsible compass to construct an equilateral triangle. This was Euclid’s Proposition I. (Propositions are theorems, mathematical statements that have to be proved.)
Proposition II is equally beautiful and elegant, even though a tiny bit less straightforward. It tells us that, given a segment BC and point A not on BC, we can draw a segment from point A such that this new segment is congruent to segment BC.
Remember, your ruler has no markings, and you are not allowed to transfer distances with neither the compass, nor the ruler.
First, connect A and B to get segment AB. Using Proposition I, built equilateral triangle DBA (for simplicity reasons, the two circles used to construct triangle DBA are not shown here).
Next, draw circle C1 with center at B and radius BC. Extend segment DB such that Circle C1 intersects DB at point E.
Last, draw circle C2 with center at D and radius DE.
Circle C2 intersects the extended DA segment at F.
Based on the definition of circles, we now have a couple of congruences:
DE ≡ DF, (1)
DB ≡ DA. (2)
In Common Notion 2 Euclid tells us that “if equals are subtracted from equals, then the remainders are equal.” If we subtract (2) from (1) we have
BE ≡ AF. (3)
But BE and BC are radii of the same circle, therefore
BE ≡ BC. (4)
From (3) and (4) we have AF ≡ BC.
Q.E.F. (Quod erat faciendum, or That which was to be done, as Euclid would usually end a construction)