ABCD is a parallelogram. E and F are the midpoints of AB and CD respectively. GM is any line intersecting AD, EF and BC at G, P and H respectively. Prove that GP = PH

Asked by Paresh | 11th Dec, 2015, 02:56: PM

Expert Answer:

C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
A B C D space i s space a space p a r a l l e log r a m comma space E space a n d space F space a r e space t h e space m i d p o i n t s space o f space t h e space s i d e s space A B space a n d space C D space r e s p e c t i v e l y. T h e r e f o r e comma space A D parallel to E F parallel to B C T h e space l i n e space G M space i n t e r s e c t s space t h e space l i n e space s e g m e n t s space A D comma space E F space a n d space B C space a t space G comma space P space a n d space H space r e s p e c t i v e l y. W e space n e e d space t o space p r o v e space t h a t space G P equals P H I f space t h r e e space o r space m o r e space p a r a l l e l space l i n e s space i n t e r s e c t space a space t r a n s v e r s a l comma space t h e n space t h e y space c u t space o f f space t h e space t r a n s v e r s a l space p r o p o r t i o n a l l y. H e r e comma space A D comma space E F space a n d space B C space a r e space t h r e e space p a r a l l e l space l i n e s space a n d space G H space i s space t h e space t r a n s v e r s a l.  therefore fraction numerator A E over denominator E B end fraction equals fraction numerator C F over denominator D F end fraction equals fraction numerator G P over denominator P H end fraction S i n c e space E space a n d space F space a r e space m i d p o i n t s space o f space A B space a n d space C D space r e s p e c t i v e l y comma space w e space h a v e comma A E equals E B space a n d space C F equals D F rightwards double arrow fraction numerator G P over denominator P H end fraction equals 1 rightwards double arrow G P equals P H

Answered by Vimala Ramamurthy | 12th Dec, 2015, 12:49: PM