---
book: 11
number: 26
id: "XI.26"
kind: "construction"
uses: ["[[book-11/proposition-11]]", "[[book-1/proposition-23]]", "[[book-11/proposition-12]]", "[[book-1/proposition-4]]", "[[book-1/proposition-8]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.26"
license: "CC-BY-SA-4.0"
---

# XI.26

*On a given straight line, and at a given point on it*, *to construct a solid angle equal to a given solid angle.*

## Proof

Let *AB* be the given straight line, *A* the given point on it, and the angle at *D*, contained by the angles *EDC*, *EDF*, *FDC*, the given solid angle; thus it is required to construct on the straight line *AB*, and at the point *A* on it, a solid angle equal to the solid angle at *D*.

For let a point *F* be taken at random on *DF*, let *FG* be drawn from *F* perpendicular to the plane through *ED*, *DC*, and let it meet the plane at *G*, [[book-11/proposition-11|XI. 11]] let *DG* be joined, let there be constructed on the straight line *AB* and at the point *A* on it the angle *BAL* equal to the angle *EDC*, and the angle *BAK* equal to the angle *EDG*, [[book-1/proposition-23|I. 23]] let *AK* be made equal to *DG*, let *KH* be set up from the point *K* at right angles to the plane through *BA*, *AL*, [[book-11/proposition-12|XI. 12]] let *KH* be made equal to *GF*, and let *HA* be joined; I say that the solid angle at *A*, contained by the angles *BAL*, *BAH*, *HAL* is equal to the solid angle at *D* contained by the angles *EDC*, *EDF*, *FDC*.

For let *AB*, *DE* be cut off equal to one another, and let *HB*, *KB*, *FE*, *GE* be joined.

Then, since *FG* is at right angles to the plane of reference, it will also make right angles with all the straight lines which meet it and are in the plane of reference; [[book-11/definitions#Definition 3|XI. Def. 3]] therefore each of the angles *FGD*, *FGE* is right.

For the same reason each of the angles *HKA*, *HKB* is also right.

And, since the two sides *KA*, *AB* are equal to the two sides *GD*, *DE* respectively, and they contain equal angles, therefore the base *KB* is equal to the base *GE*. [[book-1/proposition-4|I. 4]]

But *KH* is also equal to *GF*, and they contain right angles; therefore *HB* is also equal to *FE*. [[book-1/proposition-4|I. 4]]

Again, since the two sides *AK*, *KH* are equal to the two sides *DG*, *GF*, and they contain right angles, therefore the base *AH* is equal to the base *FD*. [[book-1/proposition-4|I. 4]]

But *AB* is also equal to *DE*; therefore the two sides *HA*, *AB* are equal to the two sides *DF*, *DE*.

And the base *HB* is equal to the base *FE*; therefore the angle *BAH* is equal to the angle *EDF*. [[book-1/proposition-8|I. 8]]

For the same reason the angle *HAL* is also equal to the angle *FDC*.

And the angle *BAL* is also equal to the angle *EDC*.

Therefore on the straight line *AB*, and at the point *A* on it, a solid angle has been constructed equal to the given solid angle at *D*. Q. E. F.