Question

If a line drawn parallel to one side of a triangle it intersect the other two sides in distinct points that the other two sides are divided in the same ratio solution

Answer 1

let a triangle ABC
D is a point on AB and E is a point on AC
DE parallel to BC
construction: draw EF perpendicular to AB, DG perpendicular to AC,join BE and CD
Proof:
area(ADE)/area(BDE)=1/2*AD*EF/1/2*BD*EF
So,area(ADE)/area(BDE)=AD/BD. (1)
similarly,
area(ADE)/area(CDE)=1/2*AE*DG/1/2*CE*DG
So,area(ADE)/area(CDE)=AE/CE. (2)
triangle BDE and triangle CDE lie on a common base and between same parallel lines
S,area(BDE)=area(CDE). (3)
from (1),(2)and (3)
Ad/BD=AE/CE