Problem
GEO-B1-M02-P020 Segments Through a Midpoint
#20
★★★☆☆ Level 3 of 5
Point \(M\) is the midpoint of segment \(AB\). Through \(M\), a line is drawn; on opposite sides of \(M\), points \(C\) and \(D\) are chosen so that \(MC=MD\). Prove that \(AC=BD\) and \(AD=BC\).
Use vertical angles at the intersection of lines \(AB\) and \(CD\).
Since \(M\) is the midpoint of \(AB\), \(AM=BM\). Also \(MC=MD\). Angles \(AMC\) and \(BMD\) are vertical, so they are equal. By SAS, \(\triangle AMC=\triangle BMD\), hence \(AC=BD\). Similarly, \(\angle AMD=\angle BMC\), and by SAS \(\triangle AMD=\triangle BMC\), so \(AD=BC\).
A very good problem with two pairs of hidden congruent triangles.