Given a triangle ABC and a point S inside.Show that if areas of triangle ABS, triangle BCS and triangle CAS are equal, then S is the centroid of ABC.
Hello,
1. The area of a triangle is calculated by:
. That means:
and
. Thus
2. A parallel to the side of a triangle through the centroid divides the height which is perpendicular to this side into 2 parts which have the ration 2 : 1. The part of the height which is connected with the side of the triangle is therefore of the complete height.
3. All these consideration are the same for all three triangles. Thus S must be the centroid.