Euclidean Construction of a Square Root
Drag the point A so that the length DA is the number whose square root you want.
The length of the vertical segment CD is its square root.
Call the number whose square root you want N.
How to do the construction with straightedge and compass or geometry software:
1. On a ray, mark off a distance 1 (BD), then a further distance DA=N.
2. Find the midpoint of BA, and make a circle whose diameter is BA.
3. Construct a perpendicular to BA at D. It intersects the circle at C.
The proof depends on several other ingredients. See the applets
Triangles Inscribed in a Circle
Three Similar Right Triangles
The Geometric Mean