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

# XI.12

*To set up a straight line at right angles to a given plane from a given point in it.*

## Proof

Let the plane of reference be the given plane, and *A* the point in it; thus it is required to set up from the point *A* a straight line at right angles to the plane of reference.

Let any elevated point *B* be conceived, from *B* let *BC* be drawn perpendicular to the plane of reference, [[book-11/proposition-11|XI. 11]] and through the point *A* let *AD* be drawn parallel to *BC*. [[book-1/proposition-31|I. 31]]

Then, since *AD*, *CB* are two parallel straight lines, while one of them, *BC*, is at right angles to the plane of reference, therefore the remaining one, *AD*, is also at right angles to the plane of reference. [[book-11/proposition-8|XI. 8]]

Therefore *AD* has been set up at right angles to the given plane from the point *A* in it.