Problem
GEO-B1-M06-P019 Equal Areas Give a Median
#19
★★★☆☆ Level 3 of 5
Point \(P\) lies inside triangle \(ABC\). It is known that \(S_{PAB}=S_{PAC}\). Prove that line \(AP\) passes through the midpoint of \(BC\).
Triangles \(PAB\) and \(PAC\) have common base \(AP\).
Triangles \(PAB\) and \(PAC\) have common base \(AP\). Equality of their areas implies that the distances from \(B\) and \(C\) to line \(AP\) are equal. Since \(B\) and \(C\) lie on opposite sides of \(AP\), line \(AP\) passes through the midpoint of \(BC\). Thus \(AP\) is a median.
This is the converse move to the previous problem; it is often useful in olympiads.