This theorem can be proven by placing the parallelogram on a coordinate plane. For simplicity, vertex P will be placed at the origin and vertex S will be placed on the x-axis.

Since P lies on the origin, its coordinates are (0,0). Point S is on the x-axis, meaning its y-coordinate is 0. Let a be the x-coordinate of S. Furthermore, let b and c be the coordinates of Q. P(0,0) Q(b,c) S(a,0) By the definition of a parallelogram and by the Parallelogram Opposite Sides Theorem, opposite sides of a parallelogram are parallel and congruent. With this information, it can be said that the coordinates of R are (a+b,c).

Knowing the coordinates of the vertices, the coordinates of the midpoint of the diagonals can be found. Use the Midpoint Formula to do so. The midpoint of QS will be calculated first.

By following the same procedure, the midpoint of PR can also be found.

M(x_1+x_2/2,y_1+y_2/2)
Diagonal	Endpoints	Substitute	Simplify
QS	Q( b, c) and S( a, 0)	M_(QS)(b+ a/2,c+ 0/2)	M_(QS)(a+b/2,c/2)
PR	P( 0, 0) and R( a+b, c)	M_(PR)(0+ a+b/2,0+ c/2)	M_(PR)(a+b/2,c/2)

The midpoints of the diagonals are the same. Therefore, the diagonals intersect at their midpoints. By the definition of a midpoint, it can be stated that PM=RM and QM=SM. Finally, by the definition of congruent segments it can be said that PM and RM are congruent, and that QM and SM are also congruent.

Therefore, the diagonals of a parallelogram bisect each other.

This theorem can be proven by using congruent triangles. Consider the parallelogram PQRS and its diagonals PR and QS. Let M be the point intersection of the diagonals.

Since PQ and SR are parallel, by the Alternate Interior Angles Theorem it can be stated that ∠ QPR ≅ ∠ SRP and that ∠ PQS ≅ ∠ RSQ. Furthermore, by the Parallelogram Opposite Sides Theorem it can be said that PQ≅SR.

Here, two angles of △PMQ and their included side are congruent to two angles of △RMS and their included side. Therefore, by the Angle-Side-Angle Congruence Theorem △PMQ and △RMS are congruent triangles. △PMQ≅△RMS Since corresponding parts of congruent triangles are congruent, PM is congruent to RM and QM is congruent to SM.

By the definition of a segment bisector, both segments PR and QS are bisected at point M. Therefore, it has been proven that the diagonals of a parallelogram bisect each other.