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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