---
book: 11
number: 11
id: "XI.11"
kind: "construction"
uses: ["[[book-1/proposition-12]]", "[[book-1/proposition-11]]", "[[book-1/proposition-31]]", "[[book-11/proposition-4]]", "[[book-11/proposition-8]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.11"
license: "CC-BY-SA-4.0"
---

# XI.11

*From a given elevated point to draw a straight line perpendicular to a given plane.*

## Proof

Let *A* be the given elevated point, and the plane of reference the given plane; thus it is required to draw from the point *A* a straight line perpendicular to the plane of reference.

Let any straight line *BC* be drawn, at random, in the plane of reference, and let *AD* be drawn from the point *A* perpendicular to *BC*. [[book-1/proposition-12|I. 12]]

If then *AD* is also perpendicular to the plane of reference, that which was enjoined will have been done.

But, if not, let *DE* be drawn from the point *D* at right angles to *BC* and in the plane of reference, [[book-1/proposition-11|I. 11]] let *AF* be drawn from *A* perpendicular to *DE*, [[book-1/proposition-12|I. 12]] and let *GH* be drawn through the point *F* parallel to *BC*. [[book-1/proposition-31|I. 31]]

Now, since *BC* is at right angles to each of the straight lines *DA*, *DE*, therefore *BC* is also at right angles to the plane through *ED*, *DA*. [[book-11/proposition-4|XI. 4]]

And *GH* is parallel to it; but, if two straight lines be parallel, and one of them be at right angles to any plane, the remaining one will also be at right angles to the same plane; [[book-11/proposition-8|XI. 8]] therefore *GH* is also at right angles to the plane through *ED*, *DA*.

Therefore *GH* is also at right angles to all the straight lines which meet it and are in the plane through *ED*, *DA*. [[book-11/definitions#Definition 3|XI. Def. 3]]

But *AF* meets it and is in the plane through *ED*, *DA*; therefore *GH* is at right angles to *FA*, so that *FA* is also at right angles to *GH*.

But *AF* is also at right angles to *DE*; therefore *AF* is at right angles to each of the straight lines *GH*, *DE*.

But, if a straight line be set up at right angles to two straight lines which cut one another, at the point of section, it will also be at right angles to the plane through them; [[book-11/proposition-4|XI. 4]] therefore *FA* is at right angles to the plane through *ED*, *GH*.

But the plane through *ED*, *GH* is the plane of reference; therefore *AF* is at right angles to the plane of reference.

Therefore from the given elevated point *A* the straight line *AF* has been drawn perpendicular to the plane of reference. Q. E. F.