Question

In triangle ABC, X is any point on AC. If Y, Z, U and V are the middle points of AX , XC , AB and BC respectively, then prove that UY parallel to VZ and UV parallel to YZ

Answer 1

The converse of basic proportionality theorem states that if a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.

In △XAB, EF∣∣AB, therefore,

EA

XE

=

FB

XF

.......(1)

In △XBC, FG∣∣BC, therefore,

FB

XF

=

GC

XG

.......(2)

Comparing equations 1 and 2, we get

EA

XE

=

GC

XG

Now in △XCA, if we use the converse of the basic proportionality theorem then we get that EG∣∣AC.

Hence, EG∣∣AC.