Question

From a point within a triangle, line segments are drawn to the vertices. A necessary and sufficient condition that the three triangles thus formed have equal areas is that the point be:

• A. the centre of the inscribed circle
• B. the centre of the circumscribed circle
• C. such that the three angles formed at the point each be ${120}^0$
• D. the intersection of the altitudes of the triangle
• E. the intersection of the medians of the triangle

Answer 1

The correct option is E the intersection of the medians of the triangle

To divide a triangle into three triangles of equal area is to find the centroid (the point where the medians intersect). After determining the centroid (point $G$ below), construct the segments connecting the vertices to the centroid. The three triangle created are of equal area.

So from a point within a triangle, line segments are drawn to the vertices. A necessary and sufficient condition that the three triangles thus formed have equal areas is that the point be the intersection of the medians of the triangle