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Doubling a Line Segment Using a Right Angle

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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.

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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 .

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Contributed by: Izidor Hafner (September 2017)
Open content licensed under CC BY-NC-SA

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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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