Triangle and its Various Kinds of Centres
What is Triangle and its Various Kinds of Centres?
The point of intersection of the internal bisectors of the angles of a triangle. It is the centre of the inscribed circle (incircle).
Key points
- Identify and define the incentre, circumcentre, orthocentre, and centroid.
- Understand the lines that intersect to form each centre.
- Recall the key properties and distance relationships associated with each centre.
- Apply these properties to solve geometric problems involving triangles.
Common exam trap
Confusing the definitions of different centres (e.g., angle bisectors vs. perpendicular bisectors).
Definitions
- Term
Incentre
- Meaning
The point of intersection of the internal bisectors of the angles of a triangle. It is the centre of the inscribed circle (incircle).
- Term
Circumcentre
- Meaning
The point of intersection of the perpendicular bisectors of the sides of a triangle. It is the centre of the circumscribed circle (circumcircle).
- Term
Orthocentre
- Meaning
The point of intersection of the altitudes of a triangle.
- Term
Centroid
- Meaning
The point of intersection of the medians of a triangle. It divides each median in the ratio 2:1.
- Term
Median
- Meaning
A line segment joining a vertex of a triangle to the midpoint of the opposite side.
- Term
Altitude
- Meaning
A perpendicular line segment from a vertex of a triangle to the opposite side (or its extension).
Learning objectives
Identify and define the incentre, circumcentre, orthocentre, and centroid.
Understand the lines that intersect to form each centre.
Recall the key properties and distance relationships associated with each centre.
Apply these properties to solve geometric problems involving triangles.
Prerequisites
Basic properties of triangles (angles, sides, medians, altitudes, angle bisectors, perpendicular bisectors).
Coordinate geometry (if problems involve coordinates).
Common mistakes
Confusing the definitions of different centres (e.g., angle bisectors vs. perpendicular bisectors).
Incorrectly applying the 2:1 ratio for the centroid.
Assuming properties of one type of triangle (e.g., equilateral) apply to all triangles.
Calculation errors when dealing with coordinates or lengths related to centres.
Keywords
Incentre
Circumcentre
Orthocentre
Centroid
Triangle
Angle Bisector
Perpendicular Bisector
Median
Altitude
Incircle
Circumcircle
Euler Line
Practice preview
If the circumcenter of a triangle lies on one of its sides, what type of triangle is it?…
hard
What is the point of intersection of the perpendicular bisectors of the sides of a triangle called?…
easy
The orthocenter of a triangle is the point of intersection of its:…
medium