Question

Ok, I've been trying this for so long but couldn't do it. Can somebody please help? Thank you.
In the triangle ABC, BF and CE are perpendicular to AC and AB respectively. D and G are the midpoints of EF and BC respectively. Prove that GD is perpendicular to EF.

Answer 1

@ash2326

Answer 2

This is really tough, but came up with a solution after 1 hour LOL!!!!

dot a perpendicular line from G to BE and label it as H
dot a perpendicular line from G to FC and label it as I

Note that by similar triangles,
consider triangles HBG and EBC
because BG = 1/2BC,
EH = 1/2BE
HG = 1/2EC
consider triangles GIC and BFC
because BG = 1/2BC,
FI = 1/2FC
GI = 1/2BF
Thus,
EG^2 = EH^2 + HG^2
= 1/4BE^2 + 1/4EC^2
= 1/4BC^2

GF^2 = GI^2 + FI^2
= 1/4BF^2 + 1/4CF^2
= 1/4BC^2
= EG^2

Thus, GF = EG

Since GF = EG, and ED = DF, then GEF is an isosceles triangle where GD is perpendicular to EF