∑Richard Hummel Math

Geometric Constructions

Building geometric figures using only a compass and straightedge — bisecting angles, constructing perpendiculars, and inscribing polygons.

Definition

Geometric constructions are drawings made using only two tools:

  • A compass — for drawing circles and arcs, and for copying lengths
  • A straightedge — for drawing straight lines (with no markings for measuring)

The constraint is important: no rulers for measuring, no protractors for measuring angles, no calculators. You can only draw circles and straight lines, and the result must be geometrically exact.

These rules were set by the ancient Greeks, who treated constructions as a way to verify geometric truth. If you can construct something, you truly understand it.

The most common constructions:

  • Bisecting a line segment (finding the midpoint)
  • Bisecting an angle
  • Constructing a perpendicular line
  • Copying an angle or segment
Bisecting a line segment

To find the midpoint of segment AB‾\overline{AB}:

  1. Open your compass to more than half the length of ABAB.
  2. Draw an arc centred at AA, above and below the segment.
  3. Without changing the compass width, draw an arc centred at BB.
  4. The two arcs intersect at two points. Draw a line through these two points.
  5. Where this line crosses AB‾\overline{AB} is the midpoint.

The construction works because the two intersection points are equidistant from both AA and BB — they lie on the perpendicular bisector of ABAB.

Try it

Describe the steps to construct the perpendicular bisector of a segment. Explain why the result passes through the midpoint and is perpendicular to the segment.

Solution

The perpendicular bisector is exactly the construction in the example. It passes through the midpoint because any point equidistant from AA and BB lies on it, and the midpoint is equidistant from AA and BB.

It is perpendicular to AB‾\overline{AB} because the perpendicular bisector is the set of all points equidistant from AA and BB — a line by symmetry — and this line makes a 90°90° angle with ABAB by the symmetry of the two arcs about the segment.

Related concepts

Geometry· Foundations
AnglesThe measure of rotation between two rays sharing a common endpoint — the foundation of shape and direction.
Geometry· Triangles
TrianglesThree-sided polygons with rich properties connecting angles, sides, area, and similarity.
Geometry· Circles
CirclesThe set of all points equidistant from a centre, with theorems about chords, tangents, arcs, and inscribed angles.
Geometry· Foundations
Geometric ProofsFormal chains of logical deduction that establish geometric truths from axioms, definitions, and previously proven theorems.

Related reading

Geometry· Foundations
Why Euclid Still MattersHow a 2,300-year-old list of five postulates gave us the template for all mathematical proof — and what happened when someone questioned the fifth one.
Geometry· Foundations
From Flat to CurvedWhat breaks when you do geometry on a sphere instead of a flat plane — and why that matters for GPS, general relativity, and the shape of the universe.