Proving the Bisectors of a Quadrilateral Bisect Each Other

[Gib Z]

Homework Statement

A quadrilateral is given, and inside it connecting opposite sides are 2 lines which bisect the sides they connect. Prove that the bisectors bisect each other.

Homework Equations

None.

The Attempt at a Solution

Well I gave the sides lengths of 2q, 2r, 2s and 2p so that when it was bisected the lengths would be nice.

I know that the bisectors are not necessarily parallel to any side, which if they were the proof would be simple. None of the angles are necessarily equal or even related in any way other than that they all add up to 2pi.

I also realized that the bisectors are necessarily perpendicular to each other. It seems all I do is show what I can't assume! I also tried to draw out pairs of sides to a certain point to form a big triangle, because I thought I might be able to do something with similarity, but to no avail. Euclidean geometry is obviously not my forte.

 

[HallsofIvy]

Is this just a "regular" geometry course or are you allowed to use Cartesian coordinates? The way I would do this would be to set up a coordinate system so that one vertex was at (0,0) and the others at (a, 0), (b, c), and (d,e). Find the coordinates of the midpoints of the sides (easy), find the equations of the bisecting lines (only a little harder) and find the point where the lines intersect.

 

[Gib Z]

It's not really any course in particular, and teachers don't mark students down for using more advanced methods unless the teacher can not follow, so I'm sure using Cartesian coordinates will work out great, although it looks like the working might get a tiny bit messy, I'll do it now.

 

[Gib Z]

Ok well My working is well over 3 pages long now, me expression for the x value of the point of intersection is 4 lines long, and once I find that point, I have to put that into the distance formula a few times. I think they intended for a different method :(