PROBLEM: Prove that if the two diagonals in a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
MY THOUGHT PROCESS:
Better draw a picture first of all. It's a quadrilateral with diagonals. We're supposed to prove that it is a parallelogram. I will try to draw a picture that doesn't look exactly like a parallelogram; in other words a picture that is not exact.
(Why? Because often, when looking at a picture that IS drawn exactly, we say, "Well I SEE that it's a parallelogram. No need proving it." So instead I want to draw a quadrilateral that doesn't at first sight look like a parallelogram.)
So what you have is a quadrilateral with two diagonals that bisect each other. Meaning that the intersection point is a midpoint for both of the diagonals.
Well right there it sounds like some line segments will have equal lengths. And, two lines crossing always form two pairs of vertical angles... So I will have some same angles and some same line segments. Sounds like I can easily prove that there are two congruent triangles and other two congruent triangles.
But how can one get from that to proving that the lines forming the quadrilateral are parallel?
It must be the corresponding angles stuff that will work there. I will have angles with same measure, so that makes that the lines must be parallel.
Okay, the proof is ready in my mind now. Just have to write it so others can understand.
PROOF WRITTEN IN 'PARAGRAPH' FORM:
Please look at the picture. Since the diagonals are bisecting each other, the line segments marked with one little line are equal, and similarly the line segments marked with double little lines. The two angles marked with dark blue line are equal, being vertical angles. It follows from SAS congruence theorem that the two yellow triangles are congruent.
Since they are congruent, angles A and A' have the same measure. And, angles A' and A'' are the same because they are vertical angles. So since A and A' are the same, and A' and A'' are the same, it follows that angles A and A'' are the same.
But this is equivalent to the two lines that form the top and bottom of the quadrilateral being parallel.
An identical argument using the two white triangles instead of the two yellow ones proves that the two sides of the quadrilateral are parallel.
So the quadrilateral is a parallelogram.