Three medians intersect at a single internal point that divides each median in segments in length ratio 2 : 1, mechanism and proof Three times the sum of squares of three sides of the triangle is equal to four times the sum of squares of the length of the medians.The perimeter of the triangle will always be larger than the sum of lengths of the three medians.This follows from the previous property and its proof will follow automatically.The sum of length of two adjacent sides of the triangle is always greater than the length of the median from the common vertex of the two sides.This property needs proof that we will discuss here. The centroid divides each median into two segments of length ratio 2 : 1, the longer side being towards the vertex.Briefly it means, if you consider a triangle as a thin metal plate of absolutely uniform thickness and density, you can balance the triangular metal plate horizontally on a vertically held pencil with the centroid placed on the tip of the pencil, at least theoretically. The centroid is called the centre of gravity of the triangle.This is a property of centroid that needs proof that we will discuss here. The centroid will always lie inside the the triangle, never outside it.The three medians in a triangle intersect at a single point which is called Centroid.The name of the intersection point is centroid by definition but the single point intersection concept needs proof which we will discuss here.A median is the line segment from a vertex of a triangle to the midpoint of the opposite side.The important concepts on medians and centroid are, This point of intersection of medians lies inside the triangle and is called the Centroid. The three medians pass through or intersect with each other at a single point inside the triangle G (in our figure). In the above figure, the medians from three vertices A, B and C are respectively, AD, BE and CF where D, E and F are the midpoints of the sides BC, CA and AB respectively. In other words, a median is the side bisector passing through a vertex. The following figure shows our objects of interest, the medians.Ī median is the line segment joining a vertex of a triangle and the mid-point of the opposite side. In any discussion on geometry the first thing that we need is a geometric figure. In this session we will go through the important concepts related to medians and also the mechanisms or proofs behind the concepts. Medians form one of the most important set of components in a triangle closely tied to the triangle independent of any other geometric shape.įor example another pair of important components, the incentre and the inradius inherit all the properties of a circle to enrich the concept of a triangle, whereas the medians and their intersection point, the centroid, throw more light on the the triangle independent of any other geometric shape. This point of intersection of medians is the centroid. Properties of a centroid: Three medians of a triangle always intersect at a single point inside the triangle.Segmentation ratio of medians at centroid: Each median is segmented at centroid in the ratio of 2 : 1 with lager segment towards the vertex.Second relation between median and sides of triangle: Three times the sum of squares of the length of sides equals four times the squares of medians of a triangle.First relation between median and sides of triangle: Sum of two sides is larger than the median from the common vertex.Relation between median and sides of triangle and other properties of medians
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