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Secondary Magazine - Issue 53: Focus on

This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 03 February 2010 by ncetm_administrator
Updated on 16 February 2010 by ncetm_administrator

Secondary Magazine Issue 53

Focus on...the centre of a triangle

orange circles

Defining the centre of triangle can be challenging – there are many different ‘centres’ to choose from! The ancient Greeks discovered some of the classic centres of triangles but didn’t give a formal definition for the centre of a triangle. The online Encyclopaedia of Triangle Centers now lists over 3 500 types of centres of triangles! 

green circles

The ‘incentre’ of a triangle is the centre of the inscribed circle – that is, the circle with the largest diameter that will fit inside the circle. The centre of the incircle is the point at which the three angle bisectors of the interior angles of the triangle meet.

incentre of a triangle

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The ‘circumcentre’ of a triangle is the centre of the circumcircle – that is, the circle whose circumference goes through each vertex of the triangle.

diagram showing that if all angles in the triangle are acute then the circumcentre is inside the triangle If all angles in the triangle are acute then the circumcentre is inside the triangle.

 diagram showing that if one of the angles of the triangle is obtuse, then the circumcentre lies outside the triangle If one of the angles of the triangle is obtuse, then the circumcentre lies outside the triangle and, if the triangle has a right angle, the circumcentre will lay on the hypotenuse of that triangle.

The diameter of the circumcircle is the length of a side of the triangle divided by the sine of the angle opposite that side.

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If the sum of the distances from a point, p, to the vertices of a triangle is the minimum possible then the point is described as the Fermat (or Toricelli) point. The strategy for finding this point was posed by Fermat to the Italian physicist and mathematician Evangelista Torricelli (1608-1647). The solution, for triangles with angles less than 120°, is to construct an equilateral triangle on each side of the original triangle then join the peak of each equilateral triangle to the opposite vertex on the original triangle. The crossing point of these three lines is the Fermat Point.

diagram showing Fermat point

red circles

The centroid of a triangle is the centre of gravity (assuming that the triangle is constructed from a solid, uniform sheet). It’s ⅓ of the distance across the triangle from each of its edges. The centroid is found by joining the midpoint of each side of the triangle with the opposite vertex. The coordinates of the centroid is the mean of the vertices of the coordinates of the vertices (that is, if the coordinates of the vertices are (1, 5), (2, 4) and (4, 7) then the x coordinate of the centroid will be found at ⅓(1+2+4) and the y coordinate will be at ⅓(5+2+7)).

diagram showing that the centre of a triangle is the centre of gravity

orange circles

The orthocentre of a triangle is the point at which the three altitudes meet (the altitude of a triangle is the perpendicular distance from one side to the opposite vertex). The orthocentre will lie within the triangle if, and only if, the all interior angles of the triangle are less than 90°. 

diagram showing the orthocentre of a triangle

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The Euler line is the straight line passing through several important points in a given triangle. It passes through the orthocenter, the circumcenter, the centroid, and the center of the nine-point circle of the triangle. In an equilateral triangle these four points coincide.


purple circles

The nine-point circle of a triangle is a circle which passes through nine significant points:

  • the three midpoints of the sides of the triangle
  • the foot of each of the three altitudes
  • the midpoint of the line segments from each of the three vertices of the triangle to the orthocenter.
The nine point centre (the centre of the nine point circle) can be found at the midpoint of the orthocentre and the circumcentre

Created with GeoGebra

This interactive diagram requires you to have Java installed. If you cannot see the diagram, you will need to install Java to be able view it. Java software is available at no cost from http://java.com/.

If you can see the diagram but it doesn't appear to work, your copy of Java may be out of date. You can download the current version of Java from http://java.com/.
Please contact us if you need further assistance.

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