Problem: 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:
Answer Choices:
A. the center of the inscribed circle
B. the center of the circumscribed circle
C. such that the three angles formed at the point each be
D. the intersection of the altitudes of the triangle
E. the intersection of the medians of the triangle
Solution:
Let be the distance from the point to any side, say , of the triangle. Then we want Area Area , or if is the altitude to the side , . . For this to hold for each side of the triangle, the required point must be the intersection of the medians.