Proving Quadrilateral Intersections: A Convexity Condition

[mathstudent88]

Show that for any quadrilateral(convex or not), the lines containing the diagonals intersect.

How would I prove this?
I know that

1) A and B are on the same side of the line CD
2) B and C are on the same side of the line DA
3) C and D are on the same side of the line AB
4) D and A are on the same side of the line BC

With this being said, is this a good proof?

Let ABCD be a quadrilateral. By condition 1 and 2 from above we know that B is in the interior of angle ADC and the ray DB intersects AC at point P. Also that A is in the interior of angle BCD and that ray CA intersects line BD at point Q. Since line DB intersect AC at point P and also at point Q, P=Q. Since P lies on AC and Q lies on BD then AC and BD have a point in common which means that the quadrilateral satisfies all the conditions of being a convex qualdrilateral also.


Thanks for the help!

 

[mighty2000]

Your proof is a good start, but there are a few steps that could use more explanation.

Firstly, it would be helpful to define what you mean by "interior" and "exterior" of an angle. This can be done by stating that the interior of an angle is the region between the two rays that form the angle, and the exterior is the region outside of the angle. This will make it clearer why B is in the interior of angle ADC and A is in the interior of angle BCD.

Next, when you say that ray DB intersects AC at point P, it would be helpful to explain why this is the case. One way to do this is to use the fact that the diagonal BD divides the quadrilateral into two triangles, and since P lies on both the diagonal and the opposite side of one of the triangles (AC), it must be the intersection point.

Similarly, when you say that ray CA intersects line BD at point Q, it would be helpful to mention that this is because Q lies on both the diagonal and the opposite side of the other triangle (BD).

Finally, it would be good to explain why the fact that P and Q are the same point (P=Q) means that the diagonals intersect. This can be done by stating that if two lines intersect at a point, then they must also intersect at every other point on those lines. Since P and Q are both on AC and BD, and we know that the two lines intersect at P=Q, then they must also intersect at every other point on AC and BD, which includes the intersection point of the diagonals.

I hope this helps to clarify your proof. Good luck!