Problem
GEO-B1-M02-P014 A Segment Inside an Isosceles Triangle
#14
★★★☆☆ Level 3 of 5
In isosceles triangle \(ABC\), \(AB=AC\). Points \(D\) and \(E\) are chosen on sides \(AB\) and \(AC\), respectively, so that \(BD=CE\). Prove that \(DE\parallel BC\).
First prove that \(AD=AE\).
Since \(AB=AC\) and \(BD=CE\), we get \(AD=AE\). Hence \(\triangle ADE\) is isosceles. Triangle \(ABC\) is also isosceles. The two triangles share the angle at \(A\), so their base angles are equal: \(\angle ADE=\angle ABC\). These are corresponding angles for lines \(DE\) and \(BC\), therefore \(DE\parallel BC\).
Contains two ideas: subtracting equal segments and transferring angles.