◀ ▲ ▶Branches / Geometry / Elements-euclid / Book-11-elementary-stereometry / Proof: By Euclid

Proof: By Euclid

(related to Proposition: Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane)

• For let some straight line $AB$ be at right angles to a reference plane.

• I say that all of the planes (passing) through $AB$ are also at "right angles to the reference plane":bookofproofs$2212.

• For let the plane $DE$ have been produced through $AB$.

• And let $CE$ be the common section of the plane $DE$ and the reference (plane).
• And let some random point $F$ have been taken on $CE$.
• And let $FG$ have been drawn from $F$, at right angles to $CE$, in the plane $DE$ [Prop. 1.11].
• And since $AB$ is at "right angles to the reference plane":bookofproofs$2212, $AB$ is thus also at right angles to all of the straight lines joined to it which are also in the reference plane [Def. 11.3] .
• Hence, it is also at right angles to $CE$.
• Thus, angle $ABF$ is a right angle.
• And $GFB$ is also a right angle.
• Thus, $AB$ is parallel to $FG$ [Prop. 1.28].
• And $AB$ is at "right angles to the reference plane":bookofproofs$2212.
• Thus, $FG$ is also at "right angles to the reference plane:bookofproofs$2212 " [Prop. 11.8] .
• And a plane is at right angles to a(nother) plane when the straight lines drawn at right angles to the common section of the planes, (and lying) in one of the planes, are at right angles to the remaining plane [Def. 11.4] .
• And $FG$, (which was) drawn at right angles to the common section of the planes, $CE$, in one of the planes, $DE$, was shown to be at "right angles to the reference plane":bookofproofs$2212.
• Thus, plane $DE$ is at right angles to the reference (plane).
• So, similarly, it can be shown that all of the planes (passing) at random through $AB$ (are) at "right angles to the reference plane":bookofproofs$2212.
• Thus, if a straight line is at right angles to some plane then all of the planes (passing) through it will also be at right angles to the same plane.
• (Which is) the very thing it was required to show.
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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"