How to prove?

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1 Answer
sankarankalyanam
Jul 7, 2018

Proved as below.

Explanation:

enter image source here

Consider triangles BEF and CDF,

#BP = CP " given"#

#hat (BPE) = hat (CPD), " Vert. opp. angles"#

# hat (PBE) = hat (DPC) " Alt. angles as " vec(BE) " // " vec (DC)#

Hence #color(crimson)(Delta BEP " congruent " Delta CDP, " ASA theorem"#

#:. bar(BE) = bar (DC)#

#color(blue)(bar (AB) = bar (DC), " opp. sides of ABCD parallelogram"#

#"Likewise", color(blue)(bar (DC) = bar (EF), " opp. sides of CDEF parallelogram"#

#"i.e. "color(green)( bar (AB) = bar (BE) = bar (EF) = bar (DC)#

#color(maroon)(bar (AF) = bar (AB) + bar (BE) + bar (EF) = 3 * bar(DC)#

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