Question

ABCD is a parallelogram In which E is the mid point of AD and O is a point on AC such that AO = 1/4 AC When EO produce meets AB at F Prove that F is the mid point of AB

Answer 1

draw the diagonal BD intersect AC at G
so, AG=GC (diagonals of a parallelogram bisect each other
now,in triangle AGD
E is a midpoint of AD and I is the midpoint of AG (AO=1/4AC and AG=GC)
so by midpoint theorem
EO parallel to BD
or, EF is parallel to BD
now, in triangle AGB
O is a midpoint of AG and FO is parallel to BG
so by converse of midpoint of theorem
F is the midpoint of AB