유클리드 기본모음 제안 43. 평행사변형에서 대각선의 한 점을 두 평행선이 만드는 평행사변형들에 대하여

제안 43. 평행사변형에서 대각선 위의 한점을 지나고 각변에 평행한 두 직선으로 네 개의 평행사변형을 얻을 때, 대각선을 포함하지 않는 두 평행사변형은 서로 면적이 같다. 

Proposition 43 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.

 

 

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