Problem
GEO-B1-M02-P009 Equal Perimeters
#9
★★☆☆☆ Level 2 of 5
Median \(AM\) of triangle \(ABC\) divides it into two triangles with equal perimeters. Prove that \(AB=AC\).
Write the equality of perimeters of triangles \(ABM\) and \(ACM\).
Since \(AM\) is a median, \(BM=CM\). By the condition, \(AB+BM+AM=AC+CM+AM\). Cancelling equal terms \(BM=CM\) and common side \(AM\), we get \(AB=AC\).
Not every problem in the module must use congruent triangles; this is a useful algebraic complement to medians.