CDS Mathematics · Circles

The Circle Through Three Points, and Locus

Exactly one circle passes through three points that are not on a line; a locus is the path of every point that satisfies one condition.

All Circles notes CDS Mathematics strategy

Why this matters

Nine PYQs, four of them HARD. The circle through a triangle's vertices is centred where the perpendicular bisectors meet, and a right angle puts the hypotenuse on a diameter. The locus items ask which curve a moving point traces — usually a circle about a fixed centre.

Concept 1 of 2: Circumcircle of a triangle

A point equally far from

A

and

B

lies on the perpendicular bisector of

AB

. Two such bisectors meet in one point, the centre of the only circle through all three vertices.

Definition

Three non-collinear points: exactly one circle. Collinear points ( $AB + BC = AC$ ): none. Two points: infinitely many.
A point at the same distance from all three vertices is the circumcentre, and that distance is the circumradius $R$ .
Right triangle: the hypotenuse is a diameter, so $R = \dfrac{\text{hypotenuse}}{2}$ .
Any triangle: $R = \dfrac{abc}{4\Delta}$ ; equilateral with side $a$ : $R = \dfrac{a}{\sqrt3}$ .
Each side is a chord, so its distance from the circumcentre is $\sqrt{R^2 - (\text{side}/2)^2}$ .

Circumradius

R = \dfrac{abc}{4\Delta}

Worked example

A triangle with sides

7

24

and

25

cm is inscribed in a circle. Find the radius.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (II) 2022 — Elementary Mathematics · Q45Moderate

Example 1 · Circles · Circumcircle and Locus

Consider the following for the items that follow : A triangle ABC with sides AB = 15 cm, BC = 9 cm, CA = 12 cm is inscribed in a circle.

What is the radius of the circle ?

Equal distance from every vertex means circumcentre

A point 'at the same distance from each vertex' is the circumcentre, not the incentre. The circle it centres may be a different, smaller circle; the sides are then chords of the circumcircle.

Drill 5 more on circumcircle of a triangle

Concept 2 of 2: Locus of a moving point

If a condition fixes a point's distance from one fixed point, the point moves on a circle about it. Most locus questions hide that fixed distance; find it and the answer is a circle.

Definition

Midpoints of all radii of a circle of radius $r$ : a concentric circle of radius $\dfrac r2$ .
Midpoints of all chords of length $c$ : a concentric circle of radius $\sqrt{r^2 - c^2/4}$ .
$PA^2 + PB^2 =$ constant: a circle centred at the midpoint of $AB$ .
$PA = PB$ : the perpendicular bisector of $AB$ , a line.
$\angle APB = 90^\circ$ : the circle on $AB$ as diameter.

Sum of squares

PA^2 + PB^2 = 2\,PM^2 + \dfrac{AB^2}{2}

Worked example

Find the locus of the midpoints of chords

10

cm long in a circle of radius

13

cm.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q76Easy

Example 2 · Circles · Circumcircle and Locus

The locus of the mid-points of the radii of length 16 cm of a circle is

Sum of squares is a circle, not a line

Equal DISTANCES from

A

and

B

give the perpendicular bisector. A constant SUM OF SQUARES of the distances gives a circle about the midpoint; the bisector is the distractor.

Drill 2 more on locus of a moving point

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Circumcircle of a triangle
Circumradius
$R = \dfrac{abc}{4\Delta}$
Locus of a moving point
Sum of squares
$PA^2 + PB^2 = 2\,PM^2 + \dfrac{AB^2}{2}$

Watch out for (2)

Equal distance from every vertex means circumcentre
→ Circumcircle of a triangle
Sum of squares is a circle, not a line
→ Locus of a moving point

Test yourself on Circles

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Circles chapter test Drill all 9 questions