Geometric Constructions Lab

Classical constructions use only two tools. A compass draws circles and arcs of any radius. A straightedge draws lines through two existing points but has no distance markings.

You may not transfer a compass opening to a new center directly (the "collapsing compass" rule), but Euclid proved this restriction does not limit what you can construct.

Every construction works at any scale because it depends only on the relative positions of points, not on absolute distances.

The perpendicular bisector of segment AB passes through the midpoint of AB at a right angle. Every point on this line is equidistant from A and B.

The construction uses two equal-radius circles. Their intersection points determine a line that is both perpendicular to AB and passes through its midpoint.

This is the foundation for many other constructions, including finding midpoints, dropping perpendiculars, and constructing squares.

The angle bisector divides an angle into two equal halves. Every point on the bisector is equidistant from both sides of the angle.

The method uses a circle at the vertex to mark equal distances on both rays, then finds the intersection of two arcs centered at those marks. The vertex-to-intersection line bisects the angle perfectly.

Angle bisectors are important in triangle geometry. The three angle bisectors of any triangle meet at a single point called the incenter, the center of the inscribed circle.

A regular hexagon inscribed in a circle has the special property that its side length equals the circumradius. This is because the central angle subtended by each side is exactly 60°.

The construction steps the circumradius around the circle six times. Each step creates a 60° arc, and after six steps the polygon closes exactly.

Not all regular polygons can be constructed with compass and straightedge. Gauss proved that a regular n-gon is constructible if and only if n is a product of a power of 2 and distinct Fermat primes (3, 5, 17, 257, 65537).