Notebook
00:02
28 Mar 2026

The Locus Definition and the Equal-Radii Engine

Most students can draw a circle, but very few can define one precisely enough to prove anything with it.

That is what this section fixes.

First we will nail the three-part locus definition — the sentence every theorem in this chapter traces back to.

Then we will pull out the operational consequence hiding inside it:

  • All radii are equal.

From there, we will see why drawing two radii automatically gives you an isosceles triangle, and why that matters for congruence proofs.

By the end, you will understand the chain that powers the entire chapter:

  1. Definition
  2. Equal radii
  3. Isosceles triangle
  4. Congruence

Along the way, expect questions that test whether you can:

SkillGoal
DefineState the definition with nothing missing
DistinguishTell a circle apart from a disc
ExplainWhy equal radii are a proof tool, not just an observation

1. The three-part locus definition of a circle

-10-551050OCompass PathP1P2P3P4P5Compass Construction: The needle is fixed atO, and the pencil traces points at a constantdistance (radius = 3) from the center.4 cmEvery point on the circle is exactly 4 cm fromthe center O.OCompass PathP1P2P3P4P5Compass Construction: The needle is fixed atO, and the pencil traces points at a constantdistance (radius = 3) from the center.

We're trying to figure out what a circle actually is — not just as a shape you recognize, but as a precise mathematical object.

Every theorem in this chapter traces back to one definition, so we need to get that definition exactly right before anything else.

📋 Given Info

You have drawn circles with a compass:

  1. Fix the needle at one point.
  2. Hold the pencil arm at a fixed distance.
  3. Rotate the compass.

The result of this process is a circle.

Look at the figure above:

  • Point OO is where the compass needle is fixed.
  • The dashed lines show the pencil arm reaching out to various points on the circle.
✍️ Question

Your turn 🎯

Define a circle in precise mathematical terms.

What are the 'centre' and the 'radius' in this definition?

Building the Definition

Let's build the definition from scratch.

Take a point O on a blank page.

✍️ MCQ
Choose one
In the context of a circle, what do we usually call this fixed point OO?

Now mark every point that is exactly 4 cm from O.

Where are these points?

✍️ MCQ
Choose one
How many points are exactly 44 cm away from point OO?

There are infinitely many of them, and together they form a ring around O — a circle.

Adding the measurement label shows that the fixed distance in our definition isn't just an idea — it's a specific, constant value that every point on the circle shares.
-10-551050OCompass PathP1P2P3P4P5Compass Construction: The needle is fixed atO, and the pencil traces points at a constantdistance (radius = 3) from the center.4 cmEvery point on the circle is exactly 4 cm fromthe center O.OCompass PathP1P2P3P4P5Compass Construction: The needle is fixed atO, and the pencil traces points at a constantdistance (radius = 3) from the center.4 cmEvery point on the circle is exactly 4 cm fromthe center O.
✍️ FIB
Fill in the blank
If we measure the distance from OO to P3P_3, or OO to P5P_5, or any other point on the circle, what would we get?
Type your answer, or hold Space to speak

Here is the precise definition:

A circle is the collection of all points in a plane that are at a fixed distance from a fixed point.

✍️ T/F
True or False?
For a shape to be a circle, every point on its boundary must be at the exact same distance from the fixed point.

The fixed point is called the CENTRE of the circle. The fixed distance is called the RADIUS.

✍️ MCQ
Choose one
If the distance from the centre OO to a point PP on the circle is 77 cm, what is the length of the radius?

Three elements make this definition complete:

  1. 'All points' — not some, not a few, every point satisfying the condition.
  2. 'At a fixed distance' — one specific number, exactly that distance, not approximately.
  3. 'From a fixed point' — one specific point, the centre.
✍️ Yes/No
Yes or No?
If we only marked some of the points that were 44 cm away from the centre instead of all of them, would we still have a complete circle?

Drop any one of these and the definition breaks.

  • 'A round shape' describes appearance.
  • 'It has a centre and radius' names parts but doesn't state the defining relationship.

The relationship — equidistance — is what makes a circle a circle.

✍️ T/F
True or False?
The statement "A circle is a round shape with a centre" is a complete and precise mathematical definition.

The Locus Definition

This is called a LOCUS definition. A circle is not defined by how it looks, but as a set of points satisfying a distance condition.

✍️ MCQ
Choose one
A locus definition describes a shape based on:

Every theorem in this chapter traces back to this one idea.

✍️ T/F
True or False?
Every theorem in this chapter traces back to the locus definition of a circle.