The medians of a triangle intersect at the centroid, which divides each median in a 2:1 ratio. Therefore, the distances from the centroid to any vertex are equal and are two-thirds the distance from that vertex to the midpoint of the opposite side. In the given triangle,
AP = BP = CP = (AB + BC + CA) / 3
A median of a triangle is a line segment that joins a vertex of the triangle to the midpoint of the opposite side. In the image you sent, the medians of triangle ABC are AD, BE, and CF. They intersect at point P, which is called the centroid of the triangle.
Apply the property of medians
A fundamental property of medians is that they divide each other in a 2:1 ratio. This means that the distance from the centroid (P) to any vertex (A, B, or C) is two-thirds the distance from that vertex to the midpoint of the opposite side.
Solve for the unknowns
We are given that AP = BP = CP. Let x be the distance from P to any vertex (A, B, or C). Then, the distance from that vertex to the midpoint of the opposite side would be 3x.
Using the property of medians, we can set up the following equations:
AD = 2x = AB + BC + CA
BE = 2x = BA + AC + CB
CF = 2x = CA + AB + BC
Adding these three equations, we get:
6x = 2(AB + BC + CA)
Simplifying, we get:
x = (AB + BC + CA) / 3
Therefore, AP = BP = CP = x = (AB + BC + CA) / 3.