Which centre is which (NF)
A standard method from Quadrilaterals and centres of triangles: the steps and worked examples.
Steps
| Centre | Lines that meet | Special property | Memory hook |
|---|---|---|---|
| Circumcentre | perpendicular bisectors of sides | same distance to the 3 vertices; centre of the circle through them | "circle circumscribed" |
| Incentre | angle bisectors | same distance to the 3 sides; centre of the circle inside | "inside" |
| Centroid | medians (vertex to mid-point of opposite side) | balance point; divides each median 2 : 1 | "centre of mass" |
| Orthocentre | altitudes (vertex perpendicular to opposite side) | "ortho = right angle" |
Right-angled triangle: the circumcentre is the mid-point of the hypotenuse, and the orthocentre is the right-angle vertex.
- Read which lines meet: perpendicular bisectors, angle bisectors, medians or altitudes.
- Name the centre from the table.
- Use its property: equal distances to the vertices (circumcentre) or to the sides (incentre), or the 2 : 1 split of each median (centroid).
Worked examples
Example 1: Which centre is which (NF): G is where the medians of △ABC meet, and the median AD is 12 cm. Name G and find AG. Then, in a right-angled triangle with sides 6 cm, 8 cm and 10 cm, find the distance from the circumcentre to each vertex.
medians meet at the centroid, which divides each median 2 : 1
AG = 12 × 2/3 = 8 cm, GD = 4 cm
right-angled triangle: circumcentre = mid-point of the hypotenuse
distance to each vertex = 10 ÷ 2 = 5 cmFor the whole topic, read Quadrilaterals and centres of triangles: key points, common mistakes and how the DSE tests it.