Foundations of Geometry

Basic Constructions

Constructing a segment bisector, angle bisector, and perpendicular line with compass and straightedge.

Foundational20 min lesson4 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Angle Relationships

Building Exact Figures with Almost No Tools

Before reading on, think about this constraint seriously: using only a compass (which draws a circle of a fixed radius) and a straightedge (with no markings at all, so it can only draw a line through two given points, never measure a length), how would you find the exact midpoint of a segment — without a ruler to measure it?

Definition — Compass-and-Straightedge Construction

A method of drawing a precise geometric figure using only two tools: a compass, which draws all points a fixed distance from a chosen center, and an unmarked straightedge, which draws the line through two chosen points.

The trick behind finding a segment's exact midpoint uses a genuinely elegant idea: any point equidistant from both endpoints of a segment must lie on the segment's perpendicular bisector. Draw a circle centered at each endpoint, both using the same radius (larger than half the segment) — the two circles cross at two points, and both of those points are, by construction, exactly the same distance from each endpoint. The line through those two crossing points is therefore the perpendicular bisector, and it crosses the original segment at its exact midpoint.

Worked Example — Constructing a Segment's Perpendicular Bisector

Given segment AB, place the compass point on A and draw an arc with a radius greater than half of AB. Without changing the radius, place the compass on B and draw another arc, crossing the first arc at two points above and below the segment. Draw a straight line through those two intersection points — it's perpendicular to AB and passes through its exact midpoint.

Bisecting an angle uses the same equidistance idea, applied to two rays instead of two points. Drawing an arc centered at the angle's vertex marks one point on each ray at equal distance from the vertex; drawing two more equal-radius arcs centered at those two marked points finds a point equidistant from both rays — and the ray from the vertex through that point splits the original angle exactly in half.

Worked Example — Constructing an Angle Bisector

Given an angle with vertex V, draw an arc centered at V crossing both rays, marking points P and Q. From P and Q, draw two equal-radius arcs that intersect at a point R inside the angle. Ray VR bisects the original angle, since R is equidistant from both original rays by construction.

Geometry Canvas

Point mode (click to add a point)LabelsMeasurementsSnap to grid

Construct

Tool

Objects

1.
Color
P1
P1: a free point, draggable on the plane
2.
Color
P2
P2: a free point, draggable on the plane
3.
Color
P3
P3: a free point, draggable on the plane
4.
Color
poly1
poly1: the polygon through P1, P2, P3

Measurements

poly1area = 15perimeter = 17.66

Tip

Every classical construction relies on the same core idea — building points that are provably equidistant from two references, using the compass's one guaranteed property: every point on a drawn arc is exactly the same distance from its center.

Common Mistakes

Changing the compass's radius partway through a construction that requires it to stay fixed, such as using a different radius for the second arc when bisecting a segment.
Keeping the same radius for both arcs is exactly what guarantees the resulting intersection points are equidistant from both original references — changing it breaks the construction's logic entirely.
Using a ruler to measure and mark an approximate midpoint or bisector instead of completing the actual compass-and-straightedge construction.
A construction's entire purpose is exactness without measurement — an approximated 'eyeballed' midpoint isn't the same skill, even if it looks close on the page.

Key Takeaways

A compass-and-straightedge construction achieves exact results using only fixed-distance arcs and lines through two points — no measurement at all.
A segment's perpendicular bisector is found by drawing two equal-radius arcs from its endpoints and connecting where they cross.
An angle bisector uses the same equidistance idea, applied to two rays sharing a vertex.

Summary

These foundational constructions rely on provable equidistance, not approximation. With points, lines, angles, and constructions established, the next unit studies how entire figures can move around the plane without changing their size or shape.

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