Problem
GEO-B1-M02-P011 Medians in Congruent Triangles
#11
★★☆☆☆ Level 2 of 5
Triangles \(ABC\) and \(A_1B_1C_1\) are congruent. Points \(M\) and \(M_1\) are the midpoints of sides \(BC\) and \(B_1C_1\). Prove that \(AM=A_1M_1\).
Prove that triangles \(ABM\) and \(A_1B_1M_1\) are congruent.
From the congruence of the original triangles, \(AB=A_1B_1\), \(BC=B_1C_1\), and \(\angle ABC=\angle A_1B_1C_1\). Since \(M\) and \(M_1\) are midpoints, \(BM=\frac{1}{2}BC\) and \(B_1M_1=\frac{1}{2}B_1C_1\), so \(BM=B_1M_1\). By SAS, \(\triangle ABM=\triangle A_1B_1M_1\). Therefore \(AM=A_1M_1\).
This problem shows that equality of medians should be proved, not assumed.