Prove triangle GIH is isosceles... The geometry questions goes like this:
Let A, B, C and D be four distinct points such that line segments BD and AC are of equal length. let E and F be the midpoints of line segments BC and AD respectively.
Let the intersection of lines EF and BD be H, the intersection of lines EF and AC be G and the intersection of lines BD and AC be I.
Prove triangle GIHis isosceles.
I have a diagram in geogebra below (I don't know how else to present it).
The problem can definitely be solved by coordinate geometry, but I can't seem to find a solution without it.


Expanded version of my comment

ABDJ — parallelogram. AJ is equal to BD, then ACJ is isosceles triangle with AJ = AC. Then angle AJC is equal to angle ACJ.
Diagonal BJ and diagonal AD intersects in middle of AD which is F. Then F is middle of BJ. Then EF is middle line of triangle BCJ. Then EF is parallel to CJ. Then angle IGH is equal to angle ACJ.
IH is parallel to AJ and GH is parallel to CJ, then angle IHG is equal to angle AJC which is equal to angle ACJ which is equal to angle IGH. Angles IHG and IGH are equal, then triangle IGH is isosceles.
