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. | I.34 | |
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. | I.34 | |
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. | C.N.2 | |
And the whole triangle ABC also equals the whole ADC, therefore the remaining complement BK equals the remaining complement KD. | C.N.3 | |
Therefore in any parallelogram the complements of the parallelograms about the diameter equal one another. | ||
Q.E.D. |