Answer :

Here, ABCD is parallelogram.

By the properties of parallelogram,

*AD || BC* and *AB || DC*

*AD = BC* and *AB = DC*

*Also,*

*AB = AE* + *BE* and *DC = DF* + *FC*

*This means that,*

*AE = BE* = *DF = FC*

Now*, DF = AE* and DF || AE, that is *AEFD* is a parallelogram.

*Hence, AD || EF***

*Similarly,* *BEFC* is also a parallelogram.

Hence, *EF || BC*

∴ *AD || EF || BC*

Thus, *AD, EF* and *BC* are three parallel lines cut by the transversal line *DC* at *D*, *F* and *C,* respectively such that *DF* = FC.

Also, the lines *AD, EF* and *BC* are also cut by the transversal *AB* at *A, E* and *B*, respectively such that *AE* = *BE*.

Similarly, they are also cut by *GH*.

Hence by intercept theorem,

∴ GP = PH

Hence proved.

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