G.523 planes cut right

XI.19

If two planes which cut one another be at right angles to any plane, their common section is at right angles to the same plane.

planes_cut_right()

prove / B D / is at right angles to the plane of reference

assert Let the two planes < A B >, < B C > be at right angles to the plane of reference, and let / B D / be their common section

construct Suppose it is not, and from D let / D E / be drawn in the plane < A B > at right angles to / A D /, and / D F / in the plane < B C > at right angles to / C D /.

assert the plane < A B > is at right angles to the plane of reference, and / D E / has been drawn in the plane < A B > at right angles to / A D /, their common section,

∵ plane_right_angles_defassert / D E / is at right angles to the plane of reference

assert Similarly we can prove that / D F / is also at right angles to the plane of reference

∵ point_segments_setassert from the same point D two segments have been set up at right angles to the plane of reference on the same side ⇒⇐

∴ no segment except the common section / D B / of the planes < A B >, < B C > can be set up from D at right angles to the plane of reference

Dependency Graph