Triangles • Topic 4 of 4

Centres of a Triangle

Four classical points govern a triangle. The CENTROID (G) is where the three medians meet; it divides every median in a 2 : 1 ratio from the vertex and is the triangle’s centre of mass. The INCENTRE (I) is where the angle bisectors meet — the centre of the inscribed circle of radius r, and it gives the clean area link Area = r × s, where s is the semi-perimeter. The CIRCUMCENTRE (O) is where the perpendicular bisectors of the sides meet — the centre of the circle through all three vertices, of radius R = abc/(4·Area); for a right triangle O sits at the midpoint of the hypotenuse, so R = hypotenuse/2. The ORTHOCENTRE (H) is where the three altitudes meet. In any triangle G, O and H are collinear on the Euler line with HG : GO = 2 : 1. For an equilateral triangle all four centres coincide and R = 2r. CAT favours the two area relations and the right-triangle circumradius shortcut.

✅ Solved examples

1. A median of a triangle is 18 cm long. How far is the centroid from the vertex and from the midpoint of the opposite side?
The centroid divides the median 2 : 1 from the vertex. Vertex side = (2/3) × 18 = 12 cm; midpoint side = (1/3) × 18 = 6 cm.
2. A triangle has sides 13, 14, 15. Find its inradius.
s = (13+14+15)/2 = 21. Area = √[21(21−13)(21−14)(21−15)] = √[21·8·7·6] = √7056 = 84. r = Area/s = 84/21 = 4.
3. A right triangle has legs 6 and 8. Find its circumradius.
Hypotenuse = √(36+64) = 10. In a right triangle the circumcentre is the midpoint of the hypotenuse, so R = 10/2 = 5.
4. For the 13-14-15 triangle (Area 84), find the circumradius R.
R = abc/(4·Area) = (13 × 14 × 15)/(4 × 84) = 2730/336 = 8.125, i.e. R = 65/8 = 8.125.

✏️ Practice — try these, take hints as needed

1. A median is 21 cm. Distance from the centroid to the vertex?
Centroid splits the median 2 : 1 from the vertex.
Take 2/3 of the median.
2/3 × 21.
14 cm
2. A triangle has Area 30 and semi-perimeter 10. Inradius?
Area = r × s.
r = Area / s.
30 / 10.
3
3. A right triangle has hypotenuse 26. Circumradius?
Circumcentre is the midpoint of the hypotenuse.
R = hypotenuse / 2.
26 / 2.
13
4. An equilateral triangle has side 6. Inradius?
r = a / (2√3).
6 / (2√3).
Rationalise: 3/√3 = √3.
√3
5. In a triangle, HG : GO on the Euler line is 2 : 1. If GO = 4, find HO.
HG = 2 × GO.
HO = HG + GO.
8 + 4.
12

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Pythagoras Take the chapter test →