Thread: Parallelogram

In the figure, ABCD is a parallelogram.
If DX=XY=YB then

(a)AX||YC
(b)XAYC is a parallelogram
(c)Ac bisects XY
(d)All of these.

Hello,

Originally Posted by devi

In the figure, ABCD is a parallelogram.
If DX=XY=YB then

(a)AX||YC
(b)XAYC is a parallelogram
(c)Ac bisects XY
(d)All of these.

I'm only going to give you tips because it's quite a hard work proving all of these ^^
For those who don't see the sketch, D, X, Y and B are aligned on the diagonal DB.

DX=XY=YB. Prove that DX=DB/3 and BY=DB/3 (1)

(a) Prove that triangles AXB and DCY are isometric (2 equal sides and the angle inscribed equal). Then, note that 2 of their sides are parallel to each other and juggle with angles.

(b) This is a direct consequence from (a) : AX // YC and AX=YC. This is a sufficient property to prove that it's a parallelogram.

(c) Property (1) + property of the parallelogram : its diagonals intersect in their midpoint (let's call it O for parallelogram ABCD). This may be sufficient to prove that XO=YO and thus that AC bisects XY.