The circumcentre is a critical point in the geometry of a triangle. It is the point where the perpendicular bisectors of the three sides of a triangle intersect. Understanding the circumcentre is essential in solving various geometrical problems and can provide valuable insights into the properties of a triangle. In this article, we will explore the circumcentre formula and its significance in triangle geometry.
Definition of Circumcentre
The circumcentre of a triangle is the point where the perpendicular bisectors of the three sides of the triangle intersect. It is equidistant from the three vertices of the triangle, making it the center of the circumcircle that passes through all three vertices.
Significance of Circumcentre
- The circumcentre is equidistant from the vertices of the triangle.
- It lies inside the triangle if the triangle is acute-angled, outside the triangle if it is obtuse-angled, and on the triangle if it is right-angled.
- The circumcentre is the center of the circumcircle, the circle passing through all three vertices of the triangle.
- The circumcircle is the smallest circle that can contain the entire triangle within it.
Circumcentre Formula
To find the circumcentre of a triangle, we can use the following formula:
Circumcentre Formula:
Let the vertices of the triangle be A(x1, y1), B(x2, y2), and C(x3, y3).
The coordinates of the circumcentre O are given by:
[ O(x,y) = \left( \frac{(x1^2 + y1^2)(y2 – y3) + (x2^2 + y2^2)(y3 – y1) + (x3^2 + y3^2)(y1 – y2)}{2(x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2))} \ , \frac{(x1^2 + y1^2)(x3 – x2) + (x2^2 + y2^2)(x1 – x3) + (x3^2 + y3^2)(x2 – x1)}{2(x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2))} \right) ]
Derivation of Circumcentre Formula
The formula for the circumcentre can be derived using concepts from analytic geometry. By setting up the perpendicular bisectors of the sides of the triangle, we can form a system of equations that represent the perpendicular bisectors. Solving these equations simultaneously will lead us to the coordinates of the circumcentre.
Example Problem
Let’s consider a triangle with vertices A(2, 3), B(4, 6), and C(1, 5). To find the circumcentre O of this triangle, we can use the circumcentre formula:
Applying the formula, we get:
[ O(x,y) = \left( \frac{(2^2 + 3^2)(6 – 5) + (4^2 + 6^2)(5 – 3) + (1^2 + 5^2)(3 – 6)}{2(2(6 – 5) + 4(5 – 3) + 1(3 – 6))} \ , \frac{(2^2 + 3^2)(1 – 4) + (4^2 + 6^2)(2 – 1) + (1^2 + 5^2)(4 – 2)}{2(2(6 – 5) + 4(5 – 3) + 1(3 – 6))} \right) ]
After computation, we can find the coordinates of the circumcentre O which will give us the center of the circumcircle passing through the points A, B, and C.
Properties of Circumcentre
- The circumcentre is equidistant from the vertices of the triangle.
- The circumcentre lies on the perpendicular bisectors of the sides of the triangle.
- The circumcentre coincides with the centroid and orthocentre in equilateral triangles.
FAQ
Q1: What is the relationship between the circumcentre and the centroid of a triangle?
A1: In an equilateral triangle, the circumcentre, centroid, and orthocentre all coincide at the same point.
Q2: Can the circumcentre lie outside the triangle?
A2: Yes, the circumcentre can lie outside the triangle in the case of an obtuse-angled triangle.
Q3: How is the circumcentre different from the centroid and orthocentre of a triangle?
A3: The circumcentre is the point where the perpendicular bisectors of the sides of a triangle intersect, the centroid is the point where the medians intersect, and the orthocentre is the point where the altitudes intersect.
Q4: How does the circumcentre affect the circumcircle of a triangle?
A4: The circumcentre is the center of the circumcircle, the circle passing through all three vertices of the triangle.
Q5: Is there a formula to calculate the circumradius of a triangle?
A5: Yes, the circumradius (R) of a triangle can be calculated using the formula:
[ R = \frac{a}{2\sin(A)} = \frac{b}{2\sin(B)} = \frac{c}{2\sin(C)} ]
where a, b, and c are the sides of the triangle, and A, B, and C are the corresponding angles.