What are the Three Points of Concurrency in a Triangle?

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In Physics or Mathematics, the arithmetic mean of all points of a plane figure is termed as the geometric center or the centroid of that figure. In layman’s terms, we can say that if we have a cutout of that shape, the point at which this shape could be perfectly balanced at the tip of a pin is known as the centroid. The centroid of triangle is defined as the point in the triangle that is at the intersection of the medians. In this article, we learn more about the properties of the centroid and other associated concepts that can help students get a better understanding of how to solve questions on the same.
Properties of Centroid of a Triangle
We can locate the centroid of the triangle by finding the intersection point of all the medians of a triangle.
There are three points of concurrency in a triangle – incenter, centroid, and circumcenter.
The centroid is contained inside the triangle.
At the centroid, each of the three medians gets divided into a 2:1 ratio.
Formula for Centroid of a Triangle
If we have a triangle PQR and the coordinates of each vertice are given by P(x1, y1), Q(x2, y2), and R(x3, y3), then the formula is given as
Centroid = [(x1+x2+x3)/3, (y1+y2+y3)/3].
Incenter of a Triangle
The point of intersection of the angle bisectors of a triangle is known as the incenter of the triangle. The angle bisectors divide the interior angle into two equal halves. The incenter will be at an equal distance from all three sides of a triangle. It also acts as the center of the circle that is inscribed in the triangle. This point is independent of the triangle’s scale or placement.
Formula for Incenter
If the coordinates of vertices of a triangle PQR are the same as given above and the lengths of the sides are given by p, q and r, then the formula for the incenter is given as
Incenter = (px1 + qx2 + rx3p + q + r, py1 + qy2 + ry3p + q + r)
Circumcenter of a Triangle
The point where the perpendicular bisectors of the sides of a triangle intersect is known as the circumcenter of the triangle. The circumcenter acts as the center of the circumcircle of that triangle. This circle can lie inside or outside the figure.
Formula for the Circumcenter
Suppose the angles of a triangle PQR are given by P, Q, R, and the coordinates of each vertex is given as mentioned above, then the formula for the circumcenter of a triangle is as follows:
P(X, Y) = [(x1 sin 2P + x2 sin 2Q + x3 sin 2R)/ (sin 2P + sin 2Q + sin 2R), (y1 sin 2P + y2 sin 2Q + y3 sin 2R)/ (sin 2P + sin 2Q + sin 2R)]

Conclusion

There are several other concepts associated with triangles that kids must know. The best place to gain a good quality of mathematical education is Cuemath. At Cuemath, students are encouraged to follow their own learning style so that they do not get pressured. The tutors use many resources, such as online worksheets, math puzzles, workbooks, games, workbooks, apps, etc., to teach a lecture. To know more about the curriculum offered, visit the Cuemath website. Hopefully, this article has given you an insight into how to find the different points of concurrency in a triangle and helps you solve any level of question on the same.

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