What is BPT Theorm

Theorem:If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points,the other two sides are divided in the same ratio. Ans.Given:ABC is a∆ BC||DE To Prove:AD by BD=AE by EC Construction:Join DC and BE Draw ENperpendicular to AB and DM perpendicular to AE. Proof:BC||DE(given) ar.∆BDE = ar.∆DEC....................eq.1(∆on same base bet.same parallel.) ar.ADE by ar.BDE=1/2×AD×EN by 1/2×BD×EN ar.ADE by ar.BDE=AD by BD.............eq.2 ar.AED by ar.DEC=1/2×AE×DM by 1/2×EC×DM ar.AED by ar.DEC=AE by EC..............eq.3 Comparing 2 and 3 using 1 ar.ADE by ar.BDE=ar.AED by ar.DEC AD by BD=AE by EC(proved) {AD by BD+1=AE by EC+1 AD+BD by BD= AE+EC by EC AB by BD=AC by EC Taking reciprocal BD by AB= EC by AC 1-BD by AB= 1-EC by AC AB-BD by AB= AC-EC by AC AD by AB =AE by AC}corrolary.

According to this theorm ,a line is drawn parallel to the one side of a triangle to interset the other two sides in distinct points then the other two sides are divided in same ratio. If in triangle ABC , DE parallel AD then by BPT theorm AD/DB equal AE/EC