Problem
GEO-B1-M04-P017 A Diagonal as an Angle Bisector
#17
★★★☆☆ Level 3 of 5
In parallelogram \(ABCD\), diagonal \(AC\) bisects angle \(A\). Prove that \(ABCD\) is a rhombus.
Use \(AD\parallel BC\) to get one more equal angle.
By condition \(\angle BAC=\angle CAD\). Since \(AD\parallel BC\), \(\angle CAD=\angle ACB\). Therefore \(\angle BAC=\angle ACB\), so in triangle \(ABC\), sides \(AB\) and \(BC\) are equal. In a parallelogram, equality of adjacent sides means that all sides are equal. Hence \(ABCD\) is a rhombus.
A typical olympiad micro-idea: an angle bisector plus parallelism creates an isosceles triangle.