reflections on triangles

special points of a triangle

Jim Stamell, Sylvania High School

Recently my attention was drawn to four special points of any triangle, namely the centroid, orthocentre, circumcentre, and incentre. These points are usually mentioned as asides or footnotes in high school geometry texts, and rarely is anything more than how to construct them given. The following information gives some properties of these points, but I have not set out to prove the results. This may be left as an exercise for the reader. I have been unable to find any reference to a collective noun which aptly describes these four terms; hence the title of this article.

The centroid

The centroid is obtained by joining the vertices of a triangle to the midpoint of their respective opposite sides. The three medians so drawn are concurrent at the centroid, G.

This point is two-thirds distant from a vertex to the opposite side along a median. It is the centre of gravity of the triangle, or the centre of gravity of equal masses placed at the vertices. Further, if G is the centroid of the triangle PQR, then the areas of triangle PQG, triangle QRG, and triangle RPG are equal.

If the coordinates of the vertices of triangle PQR are (a, e), (b, f), and (c, g) in the Cartesian system, it may be shown that G is .

The orthocentre

This is the point of intersection of the three altitudes (the perpendiculars from the vertices to the opposite sides) of the triangle.

A pedal triangle is a triangle formed within a given triangle by joining the feet of the altitudes, so forming triangle DEF (Figure 3). The altitudes of the given triangle bisect the angles of the pedal triangle. Hence the orthocentre of triangle PQR is also the incentre of triangle DEF.

Another interesting property is the ‘nine-point theorem’, proved by Charles Brianchon and Jean Poncelet in 1820. This states that the circle that passes through the feet of the altitudes passes also through the midpoints of the segments that join the vertices to the orthocentre.

That is, PL = QL, QK = RK, and PI = RI, and also
PA = OA, QB = OB, and RC = OC.

Some years later, Feuerbach embellished this theorem by proving that not only do the nine points lie on the circle, but also the circle is tangent to the inscribed circle and to the three escribed circles of the given triangle. (An escribed, or excircle, is a circle tangent to one side of the triangle and to the extensions of the other sides.)

It can also be shown that the nine-point centre bisects the join of the circumcentre with the orthocentre of the triangle.

The circumcentre

The circumcentre is the intersection of the perpendicular bisectors of the three sides of the triangle. It is also the centre of the circumscribed circle (which passes through the vertices of the triangle).

A circumscribed circle of a polygon is a circle which passes through the vertices of the polygon. The polygon is then an inscribed polygon of the circle. If the polygon is regular, with side length s and n sides, the radius of the circle is:

It can be shown from either this formula, or first principles, that for an equilateral triangle, side s, the radius becomes

For any triangle with sides a, b, c, where the semiperimeter

then

Again, for an n-sided regular polygon, circumscribed by a circle, its area is:

and its perimeter is:

For an equilateral triangle, the corresponding formulae are

and where r is the radius of the circumscribed circle.

The incentre

The incentre is found when the bisectors of the interior angles of the triangles meet. It is also the centre of the inscribed circle (which touches the three sides of the triangle).

A polygon which circumscribes a circle (that is, the circle is an incircle or inscribed circle to the polygon) has its sides tangent to the circle. If the polygon is regular with n sides, its area is given by:

and its perimeter is:

where r is the radius of the inscribed circle. If s is the length of a side of a regular polygon, its radius is

For an equilateral triangle, these results become:

and

If tangents AB, CD and EF are drawn to the incircle of triangle PQR, such that then the sum of the radii inscribed in triangle PAB, triangle RCD, and triangle QEF is equal to the radius of the incircle of triangle PQR.

Further relations

A property linking the centroid with the circumcentre and orthocentre is Euler’s theorem. This states that in any triangle, the circumcentre, centroid, and orthocentre are collinear and that the centroid divides the segment containing the three points in the ratio 2:1.

A pattern also emerges if we consider the type of triangle. For an acute-angled triangle, all four points (orthocentre, centroid, circumcentre, and incentre) lie within the triangle. If the triangle is equilateral, the points coincide. In a right-angled triangle, two of these points lie within the triangle, the third is at the midpoint of the hypotenuse (which?), and the fourth point is at the vertex of the right angle (again which?). When the triangle is obtuse, two of these points lie outside the triangle (which?) and the other two inside.

References