Problem
GEO-B1-M02-P017 Points on Two Rays
#17
★★★☆☆ Level 3 of 5
On the sides of angle \(A\), points \(B,D\) lie on one ray and \(C,E\) on the other, with \(AB=AC\) and \(BD=CE\). Prove that \(DE\parallel BC\).
First get \(AD=AE\), then compare triangles \(ADE\) and \(ABC\) by angles.
Since the points lie on the same rays, \(AD=AB+BD\) and \(AE=AC+CE\). From \(AB=AC\) and \(BD=CE\), we get \(AD=AE\). Triangles \(ABC\) and \(ADE\) are isosceles with the same angle at \(A\), so their base angles are equal. Hence \(\angle ADE=\angle ABC\), which gives \(DE\parallel BC\).
A useful variation of Problem 14: equality is obtained by addition, not subtraction.