Solution of Triangles: Distance between various centres
Solution of Triangles: Distance between Various Centres
In the study of triangles, particularly in trigonometry, we often come across various 'centres' or 'points of concurrency' such as the centroid, circumcentre, incentre, and orthocentre. Understanding the distances between these points is crucial for solving complex geometrical problems. In this article, we will explore these centres, their properties, and the formulas to calculate the distances between them.
Centres of a Triangle
Before we delve into the distances between the centres, let's briefly define each centre:
- Centroid (G): The point where the three medians of the triangle intersect. It is also the centre of mass of the triangle.
- Circumcentre (O): The point where the perpendicular bisectors of the sides of the triangle intersect. It is the centre of the circumcircle, the circle that passes through all three vertices of the triangle.
- Incentre (I): The point where the angle bisectors of the triangle intersect. It is the centre of the incircle, the circle that is tangent to all three sides of the triangle.
- Orthocentre (H): The point where the three altitudes of the triangle intersect.
Table of Distances between Centres
| Centre 1 | Centre 2 | Distance Formula | Important Points |
|---|---|---|---|
| Centroid (G) | Circumcentre (O) | $OG = \frac{1}{3}R$ | R is the radius of the circumcircle. |
| Centroid (G) | Incentre (I) | $GI = \frac{1}{3}d$ | d is the distance between the circumcentre and the incentre. |
| Centroid (G) | Orthocentre (H) | $GH = \frac{1}{3}OH$ | OH is the distance between the orthocentre and the circumcentre. |
| Circumcentre (O) | Incentre (I) | $OI^2 = R(R - 2r)$ | R and r are the radii of the circumcircle and incircle, respectively. |
| Circumcentre (O) | Orthocentre (H) | $OH^2 = R^2 - 8R^2\cos A \cos B \cos C$ | A, B, and C are the angles at the vertices of the triangle. |
| Incentre (I) | Orthocentre (H) | No direct formula | Depends on the specific triangle. |
Formulas and Examples
Distance between Centroid and Circumcentre (OG)
The centroid divides the median in the ratio 2:1. Since the circumcentre lies on the median of the triangle, the distance between the centroid and the circumcentre is one-third the distance from the circumcentre to the midpoint of the side opposite the vertex, which is the radius of the circumcircle (R).
$$ OG = \frac{1}{3}R $$
Distance between Centroid and Incentre (GI)
The distance between the centroid and the incentre can be found using the formula:
$$ GI = \frac{1}{3}d $$
where d is the distance between the circumcentre and the incentre. This distance can be calculated using the formula for $OI$ given in the table above.
Distance between Centroid and Orthocentre (GH)
The distance between the centroid and the orthocentre is one-third the distance between the orthocentre and the circumcentre (OH).
$$ GH = \frac{1}{3}OH $$
Distance between Circumcentre and Incentre (OI)
The distance between the circumcentre and the incentre can be found using the formula:
$$ OI^2 = R(R - 2r) $$
Distance between Circumcentre and Orthocentre (OH)
The distance between the circumcentre and the orthocentre is given by:
$$ OH^2 = R^2 - 8R^2\cos A \cos B \cos C $$
Example
Consider a triangle ABC with sides a, b, and c. Let the angles at vertices A, B, and C be 60°, 60°, and 60°, respectively, making it an equilateral triangle. If the side length a = 6 units, find the distance between the circumcentre and the orthocentre.
Solution:
For an equilateral triangle, the circumcentre, centroid, and orthocentre coincide. Therefore, $OH = 0$.
However, if we were to use the formula for an arbitrary triangle, we would have:
$$ OH^2 = R^2 - 8R^2\cos A \cos B \cos C $$ $$ OH^2 = R^2 - 8R^2\left(\frac{1}{2}\right)^3 $$ $$ OH^2 = R^2 - R^2 = 0 $$
Thus, confirming that $OH = 0$ for an equilateral triangle.
Understanding these distances and their relationships is essential for solving problems in trigonometry and geometry. Remember that these formulas are derived under the assumption that the triangle is non-degenerate and the centres exist within the plane of the triangle.