Draw an angle XAY on your notebook as shown in Fig. . and on ray AX , mark points B B B B and B such that AB B B B B B B B B = cm. Similarly on ray AY , mark points C C C C and C , such that AC = C C = C C = C C = C C = cm, Join B C and BC .
Observe that AB B B AC C C and BC BC Similarly joining B C , B C and B C you see that AB B B AC C C and B C BC AB B B AC C C and B C BC AB B B AC C C and B C BC From this we observe that if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side. Therefore, we obtain the following theorem called converse of the Thales theorem. Theorem : Converse of Basic Proportionality Theorem Statement If a straight line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side. D E Fig.
. B B B B X C C C C Fig. . Proof Given : In ∆ ABC AD DB AE EC To prove : DE BC Construction : If DE is not parallel to BC , draw DF BC No.
Statement Reason . AD DB AE EC … ( ) Given . D ABC , DF BC Construction . AD DB AF FC … ( ) Thales theorem .
AE EC AF FC AE EC AF FC AE EC EC AF FC FC AC EC AC FC EC = FC Therefore, E = F Thus DE BC From ( )