What are the four centres of a triangle?

Centroid (intersection of medians), Incentre (intersection of angle bisectors), Circumcentre (intersection of perpendicular bisectors), Orthocentre (intersection of altitudes). The centroid is the easiest to compute using the average-of-vertices formula.

What is the Euler Line and why does it matter?

The Euler Line is the straight line passing through the Orthocentre (H), Centroid (G), and Circumcentre (O) in that order, with HG:GO = 2:1. This fact is tested in Paper 2 MC — if you know any two of these centres, you can find the third using section formula.

What are the Paper 2 shortcuts for right-angled triangles?

(1) Orthocentre = the vertex with the 90° angle. (2) Circumcentre = midpoint of the hypotenuse. These two shortcuts can save 3–5 minutes per question — don't solve them algebraically when you can apply these rules instantly.

Centres of Triangles

Master the Incentre, Circumcentre, Centroid, and Orthocentre. Learn the critical coordinate shortcuts and Paper 2 speed tricks to bypass long calculations.

1. The Core Concept

What are the Four Centres? Every triangle has four distinct geometric "centers," each formed by intersecting a different set of three lines (medians, angle bisectors, perpendicular bisectors, or altitudes). To score well on coordinate geometry questions, you must memorize how each centre is constructed and its unique geometric properties.

2. DSE Centres of Triangles Cheat Sheet — 4 Key Topics

Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.

Topic	Key Concept & Steps	Exam Shortcuts & Tips
1. Centroid	Medians Divides each median in a ratio of `2:1` (from vertex to base).	The easiest to calculate: `G = ((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3)`
2. Incentre	Angle Bisectors Equidistant from all three sides. Center of the inscribed circle.	Set of perpendicular distances from Incentre to the 3 lines are equal.
3. Circumcentre	Perpendicular Bisectors Equidistant from all three vertices. Center of the circumscribed circle.	Distance to all 3 vertices is equal: `PA = PB = PC = r`
4. Orthocentre	Altitudes The lines meet at right angles to the opposite sides.	Use perpendicular slopes: `m_altitude × m_base = −1`

Interactive Centres of Triangles Graphs

Centroid (形心)

Incentre (內心)

Circumcentre (外心)

Orthocentre (垂心)

💡 Exam Traps & Secrets

⚠️ The Right-Angled Triangle Shortcuts (Paper 2 Lifesavers)Orthocentre: In any right-angled triangle, the Orthocentre lies exactly on the vertex containing the 90° angle. You do not need to do any math!
Circumcentre: In any right-angled triangle, the Circumcentre is exactly the midpoint of the hypotenuse. Just apply the midpoint formula to the hypotenuse coordinates.

⚠️ The Euler LineFor any non-equilateral triangle, the Orthocentre (H), Centroid (G), and Circumcentre (O) lie on a single straight line in that exact order. Furthermore, they always maintain the ratio: HG : GO = 2 : 1.

⚠️ Special Triangles RulesIn an Equilateral Triangle, all four centres lie on the exact same coordinate point.
In an Isosceles Triangle, all four centres lie along the axis of symmetry (they are collinear).

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Basic Centroid Formula application)Find the coordinates of the centroid of a triangle with vertices A(1, 2), B(5, 6), and C(3, 8).

(Hypotenuse midpoint shortcut on a right-angled triangle)The coordinates of a triangle are P(0, 0), Q(6, 0), and R(0, 8). Find the coordinates of its circumcentre.

(Identifying the right-angle vertex at (2, −2) to bypass long calculations)Find the coordinates of the orthocentre of a triangle with vertices A(2, 6), B(2, −2), and C(8, −2).

Frequently Asked Questions

Try a Centres of Triangles DSE Practice Question