G.503 segments cut plane

XI.2

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

segments_cut_plane()

prove / A B /, / C D / are in one plane, and every triangle is in one plane

intersect Let / A B /, / C D / cut one another at E

construct segment() → F, segment() → G, segment(E, C) → / E C /, segment(E, B) → / E B /, segment(C, B) → / C B /, segment(F, G) → / F G /

construct Let / F H /, / G K / be drawn across

assert I say first that < E C B > is in one plane

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

assert A part also of one of / E C /, / E B / will be in the plane of reference, and a part in another

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

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

∵ part_segment_planeassert Which was proved absurd

assert < E C B > is in one plane

∵ part_segment_planeassert But, in whatever plane < E C B > is, in that plane also is each of / E C /, / E B /, and, in whatever plane each of / E C /, / E B / is, in that plane are / A B /, / C D / also

∴ / A B /, / C D / are in one plane, and every triangle is in one plane

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