Problem
GEO-B1-M07-P026 Choose the Construction
In triangle \(ABC\), point \(M\) is the midpoint of \(BC\). We need to prove a statement about equality of segments related to \(AB\) and \(AC\), but the diagram does not show a second equal pair of sides. What auxiliary construction is natural? State the construction and explain which congruent triangles it creates.
Extend \(AM\) beyond \(M\) by an equal segment.
The natural construction is to take point \(D\) on the extension of \(AM\) beyond \(M\) so that \(MD=AM\). Then \(AM=MD\), \(BM=MC\), and \(\angle AMB=\angle DMC\) as vertical angles. Therefore \(\triangle ABM\cong\triangle DCM\). Similarly, \(\triangle ACM\cong\triangle DBM\). Thus the construction creates two pairs of congruent triangles and parallelogram \(ABDC\).
This is a meta toolbox task: the student must name the construction and its purpose.