Understanding the Circumcentre of a Triangle
In this article, we will explore the method to find the circumcentre of a triangle given its vertices. The circumcentre of a triangle is the point where the perpendicular bisectors of the triangle's sides intersect, which is also the center of the circle that passes through all three vertices of the triangle, known as the circumcircle.
Steps to Find the Circumcentre of a Triangle
Let's consider a triangle with vertices at points A(4, 6), B(1, 2), and C(-3, 5). We'll follow a systematic approach to find the circumcentre of this triangle.
Step 1: Finding the Equation of the Circumcircle
The first step involves finding the equation of the circumcircle. We start by finding the equation of the line segment AB.
y - 6 1 - 4 2 - 6emsp;x - 4 -(3)(y - 6) -(4)(x - 4) 4x - 3y - 2 0
Using the equation of the circle with AB as its diameter, we get:
(x - 4)(x - 1) (y - 6)(y - 2) 2 - 5x 4 y2 - 8y 12 2 y2 - 5x - 8y 16 0
The equation of all circles passing through A and B can be expressed as:
x2 y2 - 5x - 8y 16 - 4x - 3y 2k 0
Including point C(-3, 5), we get:
[-32emsp;52emsp;-5emsp;-3emsp;-8emsp;5emsp;16] [4 -3 - 3 5 2] 025 - 25k 0k 1
Hence, the equation of the circumcircle is:
x2 y2 - x - 11y 18 0
Step 2: Finding the Equation of the Circumcircle with BC as Diameter
We find the equation of the circle with BC as its diameter:
(x - 1)(x - 3) (y - 2)(y - 5) 2 - 4x 3 y2 - 7y 10 2 y2 - 4x - 7y 13 0
We then intersect this with the equation of line BC:
3x - 4y 11x2 y2 - 4x - 7y 13 k(3x - 4y - 11) 0k -1x2 y2 - x - 11y 11 0
The intersection point, which is the circumcentre, is at (1/2, 11/2).
Alternative Method
Alternatively, we can use the property of an isosceles right triangle. By finding the squared sides:
a2 (4 - 1)2emsp;(6 - 2)2 25b2 (-3 - 1)2emsp;(5 - 2)2 25c2 (-3 - 4)2emsp;(5 - 6)2 50
Since a2 b2 and a2b2 c2, the triangle is isosceles right-angled. The circumcentre of an isosceles right triangle is the midpoint of the hypotenuse. Hence, the circumcentre is:
(4 - 3/2, 6 5/2) (1/2, 11/2)
Conclusion
In this article, we have explored two methods to find the circumcentre of a triangle given its vertices. The first method involves finding the equation of the circumcircle and solving it, while the second method utilizes the properties of the triangle to find the circumcentre directly. Both methods lead to the same result: the circumcentre is at (0.5, 5.5).
Related Content
If you are interested in learning more, here are some related articles:
Finding the Circumcentre of a Triangle Using Different Vertices Properties of the Circumcentre in Different Geometric Figures Formulas and Equations for Triangles