DEFG is a quadrilateral such that diagonal DF divides it into two parts of equal areas Prove that - Maths - Quadrilaterals

Question

DEFG is a quadrilateral such that diagonal DF divides it into two
parts of equal areas Prove that diagonal DF bisects GE also

Akhil Goyal · Accepted Answer

Consider the diagram below:
<img alt="" src=
"https://s3mn%20mnimgs%20com/img/shared/content_ck_images/images/DDDDD(2)%20png"
style="width: 353px; height: 233px;">

Let GA, EB pe perpendiculars on DF from G, E respectively
Given:ar(△DEF)=ar(△DGF)⇒12(DF)(GA)=12(DF)(EB)⇒GA=EB  -------------(1)Also, ∠GAO=∠EBO=90∘   -----(2)Further, ∠AOG=∠BOE (vertically opposite angles)Now, ∠AGO=180∘-∠GAO-∠AOGSimilarly, ∠BEO=180∘-∠EBO-∠BOETherefore, ∠AGO=∠BEO  ------(3)Using (1), (2), (3), we can see that △GOA≅△BOE (by ASA property)Hence, GO=EO

DEFG is a quadrilateral such that diagonal DF divides it into two parts of equal areas . Prove that diagonal DF bisects GE also .