How to prove three points are collinear when constructing a rectangle My problem is: Choose a unit segment OI. Then construct a rectangle with base 3 units and height 2 units. 
I cannot use angle measure. I know I can construct this figure from my unit segment by using a compass and straightedge. But when my construction is finished, I'm not sure how to prove that the points on the outside of the figure actually line up and are collinear. For example, how do I know that points R, I and O are collinear?

 A: You have asked a surprisingly interesting and subtle question.
You could have constructed your rectangle as @Travis suggests, by doubling $OI$ to $OR$, constructing a segment three times as long on the (horizontal) line perpendicular to $OR$, ending at the lower right corner which you might label $J$. (I haven't drawn my own figure for this answer.) Then you could construct a perpendicular to $OR$ at $R$ and a perpendicular to $OJ$ at $J$. They would intersect at the upper right hand corner of your rectangle. Then you wouldn't have had to ask your question - there's no possible noncolinearity to worry about.
What you would have to worry about (surprisingly) is whether the angle at the upper right is a right angle!
What you seem to have done is try to piece together six unit squares. The same worry raises its ugly head. A clever idea. Even if you settled the colinearity, how would you know the upper right corners of your unit squares were square?
That question has an important place in the history of geometry. A quadrilateral with two equal opposite sides each perpendicular to the a third side is called a Saccheri quadrilateral. It's easy to prove that the two other angles are equal, but you can't prove they are right angles without using the parallel postulate. And that postulate is equivalent to the theorem that the sum of the angles of a triangle is two right angles - which is the basis of the "angle measure" stuff you don't want to use. So in fact you can't construct that rectangle without implicitly using the Euclidean geometry of angles.
The wikipedia page at https://en.wikipedia.org/wiki/Saccheri_quadrilateral is a place to start reading. Searching for "Saccheri quadrilateral" finds more.
A: Presumably you constructed $R$ as the intersection of the line $\overline{OI}$ and the circle with center $I$ through $O$ (or something tantamount to this).
