G.503 segments cut plane — Heath

XI.2

If two straight lines cut one another, they are in one plane, and every triangle is in one plane.

For let the two straight lines / A B /, / C D / cut one another at the point E; I say that / A B /, / C D / are in one plane, and every triangle is in one plane.

For let points F, G be taken at random on / E C /, / E B /, let / C B /, / F G / be joined, and let / F H /, / G K / be drawn across; I say first that the triangle < E C B > is in one plane.

For, if part of the triangle < E C B >, either < F H C > or < G B K >, is in the plane of reference, and the rest in another,

a part also of one of the straight lines / E C /, / E B / will be in the plane of reference, and a part in another.

But, if the part < F C B G > of the triangle < E C B > be in the plane of reference, and the rest in another,

a part also of both the straight lines / E C /, / E B / will be in the plane of reference and a part in another:

[G.502]which was proved absurd.

Therefore the triangle < E C B > is in one plane.

But, in whatever plane the triangle < E C B > is, in that plane also is each of the straight lines / E C /, / E B /, and, in whatever plane each of the straight lines / E C /, / E B / is, in that plane are / A B /, / C D / also. [G.502]

Therefore the straight lines / A B /, / C D / are in one plane, and every triangle is in one plane.

Q. E. D.

Dependency Graph

Required for

G.508, G.510, G.511, G.512, G.513, G.514, G.518, G.522, G.528, G.530, G.531, G.532, G.541, G.542, G.543, G.544, G.545, G.546, G.547, G.549, G.550, G.554, G.555, G.556, G.557, G.558, G.559, G.560, G.563, G.565, G.570, G.571, G.572, G.573, G.577, G.578, G.592, G.594, G.597, G.600, G.601