Centres of a triangle

Skills and methods in this topic, easy to hard

1. 1The angle bisector's propertyA point on the bisector is equally far (perpendicularly) from both arms

   MethodPM = 3 cm: PN = 3 cm, ∠POM = 70° ÷ 2 = 35°

   Another way: By congruent triangles△OPM ≅ △OPN (AAS), PM = PN

2. 2The perpendicular bisector's propertyA point on the perpendicular bisector is equally far from both ends

   MethodAM = 4, PM = 3: PA = PB = 4² + 3² = 5

   Another way: By congruent triangles△PAM ≅ △PBM (SAS), PA = PB

3. 3The incentreIncentre: where the angle bisectors meet, equally far from the three sides

   Method∠IBC = 30°, ∠ICB = 40°: ∠BIC = 180° − 70° = 110°

   Another way: ∠BIC = 90° + ∠A ÷ 2∠A = 40°: 90° + 20° = 110°

4. 4The circumcentreCircumcentre: where the perpendicular bisectors meet, equally far from the three vertices

   MethodOA = OB = OC; right triangle, hypotenuse 10: OA = 10 ÷ 2 = 5

   Another way: Isosceles trianglesOA = OB: ∠OAB = ∠OBA

5. 5The centroidCentroid: where the medians meet, AG : GD = 2 : 1

   MethodAD = 12: AG = 12 × 23 = 8, GD = 4

   Another way: Three parts12 ÷ 3 = 4, AG = 2 × 4 = 8

6. 6The orthocentreOrthocentre: where the heights meet; at the right angle in a right triangle

   Method∠A = 50°: ∠ABE = 90° − 50° = 40°

   Another way: ∠BHC = 180° − ∠A360° − 90° − 90° − 50° = 130°

7. 7Telling the centres apartBisectors → incentre; ⊥ bisectors → circumcentre; medians → centroid; heights → orthocentre

   MethodRight triangle: circumcentre at the hypotenuse's middle, orthocentre at the right angle

   Another way: Read the marksRight-angle marks at the sides' middles: perpendicular bisectors, circumcentre

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1The angle bisector's property →

   In the picture, OP bisects ∠AOB, PM ⊥ OA and PN ⊥ OB. x =

   ▶Show the answer and working

   Answer: 8 cm

   P is on the bisector, so it is equally far from both arms: x = PM = 8 (∠ bisector theorem).

2. Example 2The circumcentre →

   O is the circumcentre of △ABC, and OA = 12 cm. OC =

   ▶Show the answer and working

   Answer: 12 cm

   The circumcentre is equally far from all three vertices (⊥ bisector theorem): OC = OA = 12 cm.

3. Example 3Telling the centres apart →

   Which centre of △ABC is the point P in the picture?

   • A. circumcentre
   • B. incentre
   • C. centroid
   • D. orthocentre

   Choose the right answer

   ▶Show the answer and working

   Answer: B. incentre

   See where the lines start and how they're marked: matching arcs mean angle bisectors, right angles at the sides' middles perpendicular bisectors, ticked halves medians, right angles at the far side from a vertex heights.