A construction that never measures
A construction never measures. How is that allowed? Build one and see.
Loading lab…
- Segment AB, and a compass wider than half of it.
- Circle centred at A with that radius. Postulate 3.
- Circle centred at B with the same radius. Postulate 3.
- Mark where the two circles cross.
- Line through the crossings. Postulate 1.
A chain of permissions
A construction is a drawing made with a compass and straight edge. It needs no measuring, because a postulate allows each step that draws.
You want a spot equally far from two houses, A and B. The segment AB joins them. Open the compass wider than half of AB.
Draw a circle from A. Postulate 3 allows it. Draw a circle from B with the same radius. Postulate 3 allows that too. Mark the two crossings. Draw a line through them. Postulate 1 allows that.
A step can be done only when what it needs already exists. Nothing here reads a number off a ruler.
What it gives you
Now we can state a claim. Each point on the line is equally far from A and B. The construction draws the line. The construction does not prove the claim.
Rule: A construction is a chain of steps. A postulate allows each step that draws. Marking a crossing draws nothing. It needs both circles first. The chain gives you a claim. A claim still needs a reason.