In any parallelogram the complements of the parallelograms about the diameter equal one another.

Let *ABCD* be a parallelogram, and *AC* its diameter, and about *AC* let *EH* and *FG* be parallelograms, and *BK* and *KD* the so-called complements.

I say that the complement *BK* equals the complement *KD.*

Since *ABCD* is a parallelogram, and *AC* its diameter, therefore the triangle *ABC* equals the triangle *ACD.*

Again, since *EH* is a parallelogram, and *AK* is its diameter, therefore the triangle *AEK* equals the triangle *AHK.* For the same reason the triangle *KFC* also equals *KGC.*

Now, since the triangle *AEK* equals the triangle *AHK,* and *KFC* equals *KGC,* therefore the triangle *AEK* together with *KGC* equals the triangle *AHK* together with *KFC.*

And the whole triangle *ABC* also equals the whole *ADC,* therefore the remaining complement *BK* equals the remaining complement *KD.*

Therefore *in any parallelogram the complements of the parallelograms about the diameter equal one another.*

Q.E.D.