유클리드 기본모음 제안 34. 평행사변형에서 대변과 대각에 대하여, 그리고 대각선의 면적 이등분에 대하여

제안 34. 평행사변형에서 반대편의 두 직선은 길이가 같고 반대편의 두 각들은 크기가 서로 같다. 그리고 대각선은 평행사변형의 면적을 이등분한다. 

Proposition 34 In parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas.

Let ACDB be a parallelogrammic area, and BC its diameter.

 

 

I say that the opposite sides and angles of the parallelogram ACDB equal one another, and the diameter BC bisects it.

평행사변형 ACDB의 마주보는 변과 각들의 크가가 서로 같고, 대각선 BC가 평행사변형의 크기를 이등분한다는 것을 보인다.

 

Since AB is parallel to CD, and the straight line BC falls upon them, therefore the alternate angles ABC and BCD equal one another.

 

 

Again, since AC is parallel to BD, and BC falls upon them, therefore the alternate angles ACB and CBD equal one another.

 

 

Therefore ABC and DCB are two triangles having the two angles ABC and BCA equal to the two angles DCB and CBD respectively, and one side equal to one side, namely that adjoining the equal angles and common to both of them, BC.

 

 

Therefore they also have the remaining sides equal to the remaining sides respectively, and the remaining angle to the remaining angle.

 

 

Therefore the side AB equals CD, and AC equals BD, and further the angle BAC equals the angle CDB.

 

 

Since the angle ABC equals the angle BCD, and the angle CBD equals the angle ACB, therefore the whole angle ABD equals the whole angle ACD.

 

 

And the angle BAC was also proved equal to the angle CDB.

 

 

Therefore in parallelogrammic areas the opposite sides and angles equal one another.

 

 

I say, next, that the diameter also bisects the areas.

 

 

Since AB equals CD, and BC is common, the two sides AB and BC equal the two sides DC and CB respectively, and the angle ABC equals the angle BCD. Therefore the base AC also equals DB, and the triangle ABC equals the triangle DCB.

 

 

Therefore the diameter BC bisects the parallelogram ACDB.

 

 

Therefore in parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas.

 

 

 

 

### 덧붙이는 말 ###

 

 

 

 

 

 

 

 

 

 

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