CDS Mathematics · Triangles

Centres of a Triangle

A triangle has four classical centres — centroid, incentre, circumcentre and orthocentre — each the meeting point of three special lines, and each with its own rule about where it lies.

All Triangles notes CDS Mathematics strategy

Why this matters

Sixteen PYQs. Six are statement questions about where each centre lies, which one table answers. Six more use the two radii — the circumradius of a right triangle is half the hypotenuse, its inradius is (a + b − c) ÷ 2. The hardest items come in two sets, from the 2024 (I) and 2026 (II) papers, and are solved fastest by putting the triangle on coordinates.

Concept 1 of 3: The four centres and where they lie

Each centre is where three lines of the same kind meet. Two of them — the centroid and the incentre — are built from lines that stay inside the triangle, so they are always inside. The other two move outside when the triangle has an obtuse angle.

Definition

Centroid $G$ : where the medians meet; always inside; it divides each median $2 : 1$ from the vertex.
Incentre $I$ : where the angle bisectors meet; always inside; the same distance $r$ from all three sides.
Circumcentre $O$ : where the perpendicular bisectors of the sides meet; the same distance $R$ from all three vertices, so the circumcircle is unique.
Orthocentre $H$ : where the altitudes meet. $A$ , $B$ , $C$ and $H$ form a set in which each point is the orthocentre of the triangle made by the other three.
In an equilateral triangle all four coincide.

Centre	Acute triangle	Right triangle	Obtuse triangle
Centroid	inside	inside	inside
Incentre	inside	inside	inside
Circumcentre	inside	midpoint of the hypotenuse	outside
Orthocentre	inside	at the right-angle vertex	outside In a right triangle the two legs are themselves altitudes, so they meet at the right angle.

The centroid and incentre never leave the triangle; the circumcentre and orthocentre do when an angle is obtuse.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q78Moderate

Example 1 · Triangles · Centres of a Triangle

Consider the following statements : 1. The orthocentre of a triangle always lies inside the triangle. 2. The centroid of a triangle always lies inside the triangle. 3. The orthocentre of a right angled triangle lies on the triangle. 4. The centroid of a right angled triangle lies on the triangle. Which of the above statements are correct ?

'On the triangle' is not 'inside'

The orthocentre of a right triangle is a vertex, and the circumcentre is the midpoint of a side: both lie ON the triangle. A statement saying 'inside' or 'outside' for a right triangle is false.

Drill 5 more on the four centres and where they lie

Concept 2 of 3: Circumradius and inradius

A right angle stands on a diameter, so the hypotenuse of a right triangle is the diameter of its circumcircle. The incircle touches each side, and the two tangents from a vertex are equal, which turns the inradius of a right triangle into a one-line formula.

Definition

Any triangle: $R = \dfrac{abc}{4\Delta}$ and $r = \dfrac{\Delta}{s}$ , where $s$ is half the perimeter.
Right triangle (hypotenuse $c$ ): $R = \dfrac c2$ and $r = \dfrac{a + b - c}{2}$ .
Equilateral (side $a$ ): $R = \dfrac{a}{\sqrt3}$ , $r = \dfrac{a}{2\sqrt3}$ , so $R = 2r$ .
Tangent lengths: from vertex $A$ to the incircle the tangent is $s - a$ . If the incircle touches $BC$ , $CA$ , $AB$ at $D$ , $E$ , $F$ , then $\angle EDF = 90^\circ - \dfrac A2$ .

Right-triangle radii

R = \dfrac c2, \qquad r = \dfrac{a + b - c}{2}

Worked example

Find the circumradius and the inradius of the right triangle with sides

6

8

10

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2024 · CDS (II) 2024 — Elementary Mathematics · Q94Moderate

Example 2 · Triangles · Centres of a Triangle

A circle is inscribed in a triangle

ABC

right-angled at

B

. If

AB = 5

cm and

BC = 12

cm, then what is the radius of the circle ?

Halve it

a + b - c

is the incircle's DIAMETER in a right triangle. For

8, 15, 17

it is

6

, and the radius is

3

Drill 5 more on circumradius and inradius

Concept 3 of 3: Placing the triangle on axes

The HARD centre questions ask for exact ratios along an altitude or the radius of a circle squeezed into a corner. Put the triangle on coordinates with its symmetry along an axis and each centre is the crossing of two straight lines.

Definition

Isosceles triangle: put the base on the $x$ -axis centred at the origin, the apex on the $y$ -axis. The altitude from the apex is $x = 0$ ; intersect it with one more altitude to get the orthocentre.
An altitude is perpendicular to its side: if the side has slope $m$ , the altitude has slope $-\dfrac1m$ .
A circle in a corner touching both sides of angle $A$ has its centre on the bisector of $A$ , at distance $\dfrac{\rho}{\sin(A/2)}$ from $A$ . If it also touches the incircle (radius $r$ ) from outside: $\dfrac{r - \rho}{\sin(A/2)} = r + \rho$ .

Circle in the corner of angle A

\rho = r\,\dfrac{1 - \sin(A/2)}{1 + \sin(A/2)}

Worked example

Triangle

PQR

has

QP = QR = 13

and

PR = 10

QM

is the altitude from

Q

and

H

is the orthocentre. Find

QH : HM

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (II) 2026 — Elementary Mathematics · Q90Hard

Example 3 · Triangles · Centres of a Triangle

for the items that follow : PQR is a triangle such that QP = QR = 15 cm and PR = 18 cm. PN, QM and RT are the altitudes of the triangle which intersect at O.

What is the ratio of QO to OM ?

Half the angle

The corner circle's centre is on the bisector, so the distance from the vertex is

\dfrac{\rho}{\sin(A/2)}

, not

\dfrac{\rho}{\sin A}

. Get

\sin\dfrac A2

from

\cos A

with

\sin^2\dfrac A2 = \dfrac{1 - \cos A}{2}

Drill 3 more on placing the triangle on axes

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)

Circumradius and inradius
Right-triangle radii
$R = \dfrac c2, \qquad r = \dfrac{a + b - c}{2}$
Placing the triangle on axes
Circle in the corner of angle A
$\rho = r\,\dfrac{1 - \sin(A/2)}{1 + \sin(A/2)}$

Reference tables (1)

The four centres and where they lie4 rows

Centre	Acute triangle	Right triangle	Obtuse triangle
Centroid	inside	inside	inside
Incentre	inside	inside	inside
Circumcentre	inside	midpoint of the hypotenuse	outside
Orthocentre	inside	at the right-angle vertex	outside In a right triangle the two legs are themselves altitudes, so they meet at the right angle.

The centroid and incentre never leave the triangle; the circumcentre and orthocentre do when an angle is obtuse.

Watch out for (3)

'On the triangle' is not 'inside'
→ The four centres and where they lie
Halve it
→ Circumradius and inradius
Half the angle
→ Placing the triangle on axes

Test yourself on Triangles

20 past CDS questions from this chapter, timed at 24 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 Triangles chapter test Drill all 16 questions