# Doubling a Line Segment Using a Right Angle

This Demonstration shows how to double a line segment using only a right angle (e.g. a carpenter's square, a try square or an iron square). No compass is needed.
Step 1: draw a ray
Step 2: draw a straight line through
Step 3: let be the intersection of and a perpendicular through
Step 4: let be the intersection of a perpendicular through and the extension of
Step 5: let be the intersection of the line through perpendicular to and the line through perpendicular to
Step 6: let be the perpendicular projection of to
Step 7: let be the perpendicular projection of to
Then is twice as long as .
Verification
The triangles and are congruent, and the quadrilateral is a rectangle. So .

### DETAILS

Axioms for a ruler:
The line segment between two points can be constructed.
The infinite straight line through two points can be constructed.
A half-infinite ray from a given point through another point can be constructed.
Additional axioms for a right angle:
Through a given point, a straight line perpendicular to a given straight line can be constructed.
Given a line segment and a figure , it is possible to decide whether contains a point from which subtends a right angle. If such a point exists, it is possible to construct this point.
The last axiom is not used in this Demonstration.
Reference
[1] B. I. Argunov and M. B. Balk, Elementary Geometry (in Russian), Moscow: Prosveščenie, 1966 pp. 268–269, pp. 332.

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