Question

How do I prove (step by step) Thales' Theorem?

Answer 1

Proof on Thales theorem :

If a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points then it divides the two sides in the same ratio.

Given : In ∆ABC , DE || BC and intersects AB in D and AC in E.

Prove that : AD / DB = AE / EC

Construction : Join BC,CD and draw EF ┴ BA and DG ┴ CA.

Statements

Reasons 1) EF ┴ BA 1) Construction 2) EF is the height of ∆ADE and ∆DBE 2) Definition of perpendicular 3)Area(∆ADE) = (AD .EF)/2 3)Area = (Base .height)/2 4)Area(∆DBE) =(DB.EF)/2 4) Area = (Base .height)/2 5)(Area(∆ADE))/(Area(∆DBE)) = AD/DB 5) Divide (4) by (5) 6) (Area(∆ADE))/(Area(∆DEC)) = AE/EC 6) Same as above 7) ∆DBE ~∆DEC 7) Both the ∆s are on the same base and between the same || lines. 8) Area(∆DBE)=area(∆DEC) 8) If the two triangles are similar their areas are equal 9) AD/DB =AE/EC 9) From (5) and (6) and (7)