Build a Square on a Given Side

Geometric Construction with the Compass Alone

Build a square with a given side AB.

Solution

Using Problem #4 first construct CA perpendicular to AB. Draw the circle A(B) of radius AB and center A and another centered at C and intersecting the former in points, say, M and N. We are given then circle A(B) and an arc MN that can be bisected following the construction in Problem #7. Both thus found points will lie on AC at the distance from A equal to AB. Denote one of them D: AD = AB and ∠BAD = 90°.

Swing two arcs of radius AB - one centered at D, another at B. The arcs will intersect at A and another point, say, E. The quadrilateral ADEB is a square.

Problems (Use a compass only)

1. Multiply a Line Segment by a Whole Number
2. Reflect a Point in a Line Sgement
3. Cross a Circle by a Line Segment
4. Drop a Perpendicular to a Line from a Point
5. Detect Collinearity
6. Complete a Parallelogram
7. Bisect an Arc
8. Find the Points of Intersection of a Circle with a Line Segment
9. Build a Square on a Given Side
10. Find the Fourth Proportional of Three Lengths
11. Find the Intersection of Two Straight Lines
12. Divide a Line Segment into a Whole Number of Parts
13. Find the Center of a Given Circle
14. Bisect a given line
15. Mascheroni Construction of a Regular Pentagon

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