Question

if a line through a midpoint meets another side a point will it be a midpoint

Answer 1

We have triangle ABC. D bisects AB, so AD=BD. We’re given DE is parallel to AC. We want to show E bisects BC, that is, to show BE=CE.

Proof: We extend the parallel line DE. We construct through C a line parallel to AD. The two lines meet at point F. By construction ADFC is a parallelogram.

AD=CF because they’re parallel sides of a parallelogram.

CF=BD by transitivity.

∠∠ BED = ∠∠ CEF - they’re opposite angles.

∠∠ BDE = ∠∠ CFE - this is true because BD is parallel to CF

Triangle BDE is congruent to triangle CFE - that’s angle, angle, side

BE=CE, corresponding parts of congruent triangles