Problem
GEO-B1-M08-P024 Equal Areas Give a Median
#24
★★★★☆ Level 4 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 areas means that the distances from \(B\) and \(C\) to line \(AP\) are equal. Since \(B\) and \(C\) are on opposite sides of \(AP\), line \(AP\) passes through the midpoint of \(BC\).
A strong converse idea: area determines a median.