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Properties of Triangles and Their Centres

The angles of every triangle add up to 180°. A triangle has four famous centres. Medians meet at the centroid G, which cuts each median 2 : 1. Perpendicular bisectors meet at the circumcentre O, the centre of the circle through the corners. Angle bisectors meet at the incentre I, the centre of the circle inside that touches all sides. Altitudes meet at the orthocentre H. O, G and H lie on one line, the Euler line.

🎬 Step-by-step story

  1. Here is a triangle ABC. Its three angles always add up to 180°. Drag corner C: the angles change, the total never does.
  2. A median joins a corner to the middle of the opposite side. The three medians meet at one point G, the centroid. G is two-thirds of the way down each median.
  3. A perpendicular bisector cuts a side in half at a right angle. The three meet at O, the circumcentre. O is the same distance from A, B and C, so one circle passes through all three corners.
  4. An angle bisector cuts a corner angle into two equal halves. The three meet at I, the incentre. I is the same distance from all three sides, so a circle fits snugly inside.
  5. An altitude drops from a corner to the opposite side at a right angle. The three meet at H, the orthocentre. Look: O, G and H sit on one straight line, the Euler line.
  6. Free play: move corner C. Make one angle bigger than 90° and watch O and H jump outside the triangle, while G and I always stay inside.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why do the three medians always meet at one point?

Each median cuts the triangle into two halves of equal area. Ceva's theorem shows three lines from the corners meet when the side ratios multiply to 1; for medians each ratio is 1, so they always meet.

Why is the circumcentre the same distance from all corners?

Any point on the perpendicular bisector of AB is equally far from A and B. O lies on all three bisectors, so OA = OB = OC.

What is the difference between the incircle and the circumcircle?

The circumcircle goes through the corners (outside the triangle). The incircle sits inside and touches the sides.

Can the orthocentre be outside the triangle?

Yes. In an obtuse triangle two altitudes fall outside, so H is outside. Make an angle bigger than 90° in the free play.

When are all four centres the same point?

In an equilateral triangle every median is also an altitude, angle bisector and perpendicular bisector, so G, O, I and H coincide.

Is a median the same as an altitude?

Not usually. A median goes to the midpoint; an altitude goes at 90°. They match only when the two sides from that corner are equal.

Basic properties of every triangle

A triangle has 3 sides and 3 angles. Three rules are true for every triangle:

Also, the longest side is always opposite the biggest angle.

Medians and the centroid

A median is a line from a corner (vertex) to the midpoint of the opposite side. Every triangle has three medians and they always meet at one point. We call that meeting point the centroid, G.

G divides each median in the ratio 2 : 1: the part from the corner is twice the part to the side. If the corners are (x₁, y₁), (x₂, y₂), (x₃, y₃), then G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). G is the balance point of a flat triangle, and it is always inside.

Perpendicular bisectors and the circumcentre

A perpendicular bisector of a side passes through its midpoint at 90°. Every point on it is the same distance from the two ends of that side. So where the three bisectors meet, point O, is the same distance R from A, B and C. O is the circumcentre and the circle of radius R through the corners is the circumcircle.

Where is O? Inside an acute triangle, at the midpoint of the longest side (the hypotenuse) of a right triangle, and outside an obtuse triangle.

Angle bisectors and the incentre

An angle bisector splits a corner angle into two equal parts. Every point on it is equally far from the two sides of that angle. The three bisectors meet at I, the incentre, which is the same distance r from all three sides. The circle of radius r is the incircle; it touches each side once. I is always inside.

Useful facts: r = Area ÷ s, where s = (a + b + c)/2 is half the perimeter. Angle bisector theorem: the bisector from A cuts BC into two parts in the ratio AB : AC.

Altitudes, orthocentre and the Euler line

An altitude is a line from a corner that meets the opposite side (or its extension) at 90°. The three altitudes meet at H, the orthocentre. In a right triangle H is the right-angle corner; in an obtuse triangle H is outside.

When three or more lines pass through one point we say they are concurrent. In every triangle, O, G and H lie on one straight line called the Euler line, with OG : GH = 1 : 2. In an equilateral triangle all four centres are the same point.

Ceva and Menelaus (for older students)

Ceva: take points D on BC, E on CA, F on AB. The lines AD, BE and CF meet at one point exactly when (BD/DC) × (CE/EA) × (AF/FB) = 1. This proves at once that medians are concurrent, because each ratio is 1.

Menelaus: if one straight line cuts the three sides (or their extensions) at D, E, F, then (BD/DC) × (CE/EA) × (AF/FB) = 1 using lengths (−1 with signed lengths). Ceva tests if three lines meet; Menelaus tests if three points are on one line.

Try it: find the centroid at home

Cut any triangle from thick cardboard. Fold or measure to mark the midpoints of the sides and draw the three medians with a ruler. Push a pencil tip under the point where they meet: the triangle balances! Then draw the angle bisectors (fold one side onto the next) and check they meet at a different point unless your triangle is equilateral.

Key formulas and definitions

Worked examples

1. Two angles of a triangle are 48° and 67°. Find the third.

Third = 180° − 48° − 67° = 65°.

2. Can 5 cm, 6 cm and 12 cm be the sides of a triangle?

Check the two shorter sides: 5 + 6 = 11, which is less than 12. So no triangle is possible.

3. A median AD is 12 cm long. Find AG and GD.

G divides AD as 2 : 1. AG = 12 × 2/3 = 8 cm, GD = 12 × 1/3 = 4 cm.

4. Find the centroid of the triangle with corners (1, 2), (5, 4), (3, 9).

G = ((1+5+3)/3, (2+4+9)/3) = (9/3, 15/3) = (3, 5).

5. A triangle has sides 6, 8 and 10 cm. Find the circumradius R and inradius r.

6² + 8² = 100 = 10², so it is right-angled. Area = ½ × 6 × 8 = 24. s = 12. r = 24 ÷ 12 = 2 cm. R = half the hypotenuse = 5 cm (check: abc/4A = 480/96 = 5).

6. In triangle ABC, AB = 9 cm, AC = 6 cm, BC = 10 cm. The bisector of ∠A meets BC at D. Find BD.

BD/DC = AB/AC = 9/6 = 3/2. So BD = 10 × 3/5 = 6 cm and DC = 4 cm.

7. In triangle ABC, D is on BC with BD/DC = 2, E on CA with CE/EA = 3. Where must F be on AB so that AD, BE, CF meet?

Ceva: 2 × 3 × (AF/FB) = 1, so AF/FB = 1/6. F divides AB in the ratio 1 : 6 from A.

Common mistakes

Practice quiz

1. The angles of a triangle add up to:
2. The point where the medians meet is the:
3. Which centre is equally far from all three corners?
4. In a right triangle, the orthocentre is at:
5. The centroid divides each median in the ratio:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What are the four centres of a triangle?

Centroid (medians meet), circumcentre (perpendicular bisectors meet), incentre (angle bisectors meet) and orthocentre (altitudes meet).

What is the Euler line?

The straight line through the circumcentre, centroid and orthocentre of any triangle. G sits between them with OG : GH = 1 : 2.

How do you find the centroid from coordinates?

Add the three x-values and divide by 3; do the same for the y-values.

Where this is taught

PolandLiceum ogólnokształcące, klasa IIIPlane geometry
RomaniaClasa a IX-aGeometry: Trigonometry in geometry, metric relations
Japan高校1年Properties of figures
South Korea중학교 2학년Properties of triangles and quadrilaterals
China八年级（初二）Ch.13 Triangles

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