# Prove that the vertices A,B,C,and D must all lie in the same plane. Assume the incidence axioms are valid.

Question: If the diagonals AC and BD of a quadrilateral ABCD intersect at some point E, prove that the vertices A,B,C,and D must all lie in the same plane. Assume the incidence axioms are valid.

Attempt and possible solution: I started working on this problem by looking at the definition of quadrilateral.

By definition quadrilateral " It is the set of all points on segments AB,BC,CD,and DA such that no three vertices A,B,C, and D are collinear and no two sides meet except at end points. "

Let segments AC and BC be diagonals of the quadrilateral ABCD, they intersect at E.

By the definition of quadrilateral let A, B, C lies in a same plane and they don't lie on a same line (non-collinear).

Suppose D does not lie on the same plane as A,B, and C.

But D must be adjacent to A and C.

Now consider the plane containing the line joining AD and CD.

Join the other diagonal BD.

Now BD is a line intersecting the plane containing A, B and C at B.

Also BD lies on the plane other than A,B, and C. So BD can not intersect the other diagonal at any point E other than B.

But B is not a point on the diagonal AC.

That is a contradiction to our hypothesis.

Therefore we can conclude A,B,C, and D must lie on the same plane.

• That's a trivial consequence of the fact that two lines with a point in common lie on the same plane. Commented Apr 12, 2021 at 18:15
• @Intelligentipauca So should I try another way to prove this? I was trying to prove my contradiction but also attempted by just using the definition of quadrilateral and the incidence axioms but I felt it was too simple. Commented Apr 12, 2021 at 18:36
• Just use the axiom stating that if two points lie in a plane, then the line containing these points lies in the same plane. Commented Apr 12, 2021 at 19:55
• @Intelligentipauca Thank you for your help. I will rework problem. Commented Apr 12, 2021 at 20:47

Line $$AE$$ lies on plane $$ABE$$. Line $$BE$$ lies on plane $$ABE$$. Done.