# 1957 AHSME Problems/Problem 26

## 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:

$\textbf{(A)}\ \text{the center of the inscribed circle} \qquad \\ \textbf{(B)}\ \text{the center of the circumscribed circle}\qquad\\ \textbf{(C)}\ \text{such that the three angles formed at the point each be }{120^\circ}\qquad\\ \textbf{(D)}\ \text{the intersection of the altitudes of the triangle}\qquad\\ \textbf{(E)}\ \text{the intersection of the medians of the triangle}$

## Solution

$[asy] import geometry; point A = (0,0); point B = (3,1); point C = (2,4); point P = (A+B+C)/3; draw(A--B--C--A); dot(A); label("A",A,SW); dot(B); label("B",B,SE); dot(C); label("C",C,NW); dot(P); label("P",P,E); draw(A--P); draw(B--P); draw(C--P); [/asy]$

Suppose the triangle is $\triangle ABC$ with the described point in its interior being $P$, as in the diagram. First, suppose that the smaller triangles have equal area (say $\tfrac{A}3$, where $A=[\triangle ABC]$). Then, by rearranging the area formula for a triangle, we see that the distance from $P$ to side $\overline{BC}$ is $\tfrac{2A}{3BC}$. Similarly, we can see that the distance from $A$ to $\overline{BC}$ is $\tfrac{2A}{BC}$. Thus, $P$ is $\tfrac1 3$ of the distance that $A$ is from $\overline{BC}$. We can use the same logic for points $B$ and $C$ and their respective opposite sides. The only point which is posiitioned in this way for all three points of the triangle is the centroid, so it is necessary that $P$ be the intersection of the medians.

Regarding the sufficiency of this condition, because the centroid is $\tfrac2 3$ of the way along each of the medians of the triangle, the three smaller triangles, with the same base and a third of the height of the large triangle, have one third of the area of the larger triangle. Thus, they all have equal areas, so it is sufficient that point $P$ is the centroid.

Thus, our answer is $\fbox{\textbf{(E) }the intersection of the medians of the triangle}$.

To disprove the other answers, try to draw counterexamples in extreme cases (so that it is obvious that a specific answer choice is incorrect).