유클리드 기본원론 제안 39. 밑변을 공유하는 같은 방향의 같은 면적의 삼각형의 정점들은 밑변에 평행한 한 직선위에 있다

제안 39. 같은 밑변을 가지고 같은 쪽에 있는 같은 면적의 삼각형은 평행한 두 직선에 있다.

Proposition 39 Equal triangles which are on the same base and on the same side are also in the same parallels.

 

 

Let ABC and DBC be equal triangles which are on the same base BC and on the same side of it. Join AD.

I say that AD is parallel to BC.

If not, draw AE through the point A parallel to the straight line BC, and join EC.

Therefore the triangle ABC equals the triangle EBC, for it is on the same base BC with it and in the same parallels.

But ABC equals DBC, therefore DBC also equals EBC, the greater equals the less, which is impossible.

Therefore AE is not parallel to BC.

Similarly we can prove that neither is any other straight line except AD, therefore AD is parallel to BC.

Therefore equal triangles which are on the same base and on the same side are also in the same parallels.

 

 

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