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C.2 Construction Theorems

Compass and straightedge constructions form an important part of the Greek’s approach to geometry and proofs. The Greeks transformed compass and straightedge constructions into calculation tools, demostrations of complex ideas and even formal proofs.

Theorems

  1. Radii of congruent circles are congruent.

  2. Consider two circles \(p\) and \(q\) with centers \(P\) and \(Q\) respectively. If \(p\) and \(q\) intersect in exactly two points, say, \(R\) and \(S\), then quadrilateral \(PRQS\) is a kite. If \(p\cong q\), \(PQRS\) is a rhombus. If \(p\in Q\) and \(q\in P\), \(\triangle{PRQ}\) is equilateral, as is \(\triangle{PSQ}\).

Basic Constructions

We can construct the following using only a compass and straightedge and, using the first and second construction theorems, prove the constructions have the required properties.

  1. Construct a segment of given length.

  2. Find the midpoint of given segment.

  3. Construct the perpendicular bisector of a segement.

  4. Construct the bisector of an angle.

  5. Construct a right triangle.

  6. Construct an isoceles triangle.

  7. Construct a 30-60-90 right triangle.

  8. Construct a 45-45-90 right triangle.

  9. Construct and equilateral triange.

  10. Construct a square.

C.1 Pathways for Constructions Axioms for Euclidean Geometry