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The triangle medians and the centroid.. In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. . Every triangle has exactly three medians, one from each vertex, and they all intersect at the triangle's cent
For 3 (non-collinear) points, if any angle of the triangle formed by those points is 120° or more, then the geometric median is the point at the vertex of that angle. If all the angles are less than 120°, the geometric median is the point inside the triangle which subtends an angle of 120° to each three pairs of triangle vertices. [10]
In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side.
Let a, b, c be the lengths of the sides of a triangle. Let d be the length of a cevian to the side of length a . If the cevian divides the side of length a into two segments of length m and n , with m adjacent to c and n adjacent to b , then Stewart's theorem states that b 2 m + c 2 n = a ( d 2 + m n ) . {\displaystyle b^{2}m+c^{2}n=a(d^{2}+mn).}
The median triangle of a given (reference) triangle is a triangle, the sides of which are equal and parallel to the medians of its reference triangle. The area of the median triangle is of the area of its reference triangle, and the median triangle of the median triangle is similar to the reference triangle of the first median triangle with a ...
A triangle's centroid is the point that maximizes the product of the directed distances of a point from the triangle's sidelines. [ 20 ] Let A B C {\displaystyle ABC} be a triangle, let G {\displaystyle G} be its centroid, and let D , E , F {\displaystyle D,E,F} be the midpoints of segments B C , C A , A B , {\displaystyle BC,CA,AB,} respectively.
The area formula for a triangle can be proven by cutting two copies of the triangle into pieces and rearranging them into a rectangle. In the Euclidean plane, area is defined by comparison with a square of side length , which has area 1. There are several ways to calculate the area of an arbitrary triangle.
The square of twice any median of an integer triangle is an integer, because the general formula for the squared median m a 2 to side a is (+), giving (2m a) 2 = 2b 2 + 2c 2 − a 2 (and likewise for the medians to the other sides).