---
book: 12
number: 3
id: "XII.3"
kind: "theorem"
uses: ["[[book-6/proposition-2]]", "[[book-1/proposition-34]]", "[[book-1/proposition-4]]", "[[book-11/proposition-10]]", "[[book-1/proposition-29]]", "[[book-11/proposition-39]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:12.prop.3"
license: "CC-BY-SA-4.0"
---

# XII.3

*Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another*, *similar to the whole and having triangular bases*, *and into two equal prisms; and the two prisms are greater than the half of the whole pyramid.*

## Proof

Let there be a pyramid of which the triangle *ABC* is the base and the point *D* the vertex; I say that the pyramid *ABCD* is divided into two pyramids equal to one another, having triangular bases and similar to the whole pyramid, and into two equal prisms; and the two prisms are greater than the half of the whole pyramid.

For let *AB*, *BC*, *CA*, *AD*, *DB*, *DC* be bisected at the points *E*, *F*, *G*, *H*, *K*, *L*, and let *HE*, *EG*, *GH*, *HK*, *KL*, *LH*, *KF*, *FG* be joined.

Since *AE* is equal to *EB*, and *AH* to *DH*, therefore *EH* is parallel to *DB*. [[book-6/proposition-2|VI. 2]]

For the same reason *HK* is also parallel to *AB*.

Therefore *HEBK* is a parallelogram; therefore *HK* is equal to *EB*. [[book-1/proposition-34|I. 34]]

But *EB* is equal to *EA*; therefore *AE* is also equal to *HK*.

But *AH* is also equal to *HD*; therefore the two sides *EA*, *AH* are equal to the two sides *KH*, *HD* respectively, and the angle *EAH* is equal to the angle *KHD*; therefore the base *EH* is equal to the base *KD*. [[book-1/proposition-4|I. 4]]

Therefore the triangle *AEH* is equal and similar to the triangle *HKD*.

For the same reason the triangle *AHG* is also equal and similar to the triangle *HLD*.

Now, since two straight lines *EH*, *HG* meeting one another are parallel to two straight lines *KD*, *DL* meeting one another, and are not in the same plane, they will contain equal angles. [[book-11/proposition-10|XI. 10]]

Therefore the angle *EHG* is equal to the angle *KDL*.

And, since the two straight lines *EH*, *HG* are equal to the two *KD*, *DL* respectively, and the angle *EHG* is equal to the angle *KDL*, therefore the base *EG* is equal to the base *KL*; [[book-1/proposition-4|I. 4]] therefore the triangle *EHG* is equal and similar to the triangle *KDL*.

For the same reason the triangle *AEG* is also equal and similar to the triangle *HKL*.

Therefore the pyramid of which the triangle *AEG* is the base and the point *H* the vertex is equal and similar to the pyramid of which the triangle *HKL* is the base and the point *D* the vertex. [[book-11/definitions#Definition 10|XI. Def. 10]]

And, since *HK* has been drawn parallel to *AB*, one of the sides of the triangle *ADB*, the triangle *ADB* is equiangular to the triangle *DHK*, [[book-1/proposition-29|I. 29]] and they have their sides proportional; therefore the triangle *ADB* is similar to the triangle *DHK*. [[book-6/definitions#Definition 1|VI. Def. 1]]

For the same reason the triangle *DBC* is also similar to the triangle *DKL*, and the triangle *ADC* to the triangle *DLH*.

Now, since the two straight lines *BA*, *AC* meeting one another are parallel to the two straight lines *KH*, *HL* meeting one another, not in the same plane, they will contain equal angles. [[book-11/proposition-10|XI. 10]]

Therefore the angle *BAC* is equal to the angle *KHL*.

And, as *BA* is to *AC*, so is *KH* to *HL*; therefore the triangle *ABC* is similar to the triangle *HKL*.

Therefore also the pyramid of which the triangle *ABC* is the base and the point *D* the vertex is similar to the pyramid of which the triangle *HKL* is the base and the point *D* the vertex.

But the pyramid of which the triangle *HKL* is the base and the point *D* the vertex was proved similar to the pyramid of which the triangle *AEG* is the base and the point *H* the vertex.

Therefore each of the pyramids *AEGH*, *HKLD* is similar to the whole pyramid *ABCD*.

Next, since *BF* is equal to *FC*, the parallelogram *EBFG* is double of the triangle *GFC*.

And since, if there be two prisms of equal height, and one have a parallelogram as base, and the other a triangle, and if the parallelogram be double of the triangle, the prisms are equal, [[book-11/proposition-39|XI. 39]] therefore the prism contained by the two triangles *BKF*, *EHG*, and the three parallelograms *EBFG*, *EBKH*, *HKFG* is equal to the prism contained by the two triangles *GFC*, *HKL* and the three parallelograms *KFCL*, *LCGH*, *HKFG*.

And it is manifest that each of the prisms, namely that in which the parallelogram *EBFG* is the base and the straight line *HK* is its opposite, and that in which the triangle *GFC* is the base and the triangle *HKL* its opposite, is greater than each of the pyramids of which the triangles *AEG*, *HKL* are the bases and the points *H*, *D* the vertices, inasmuch as, if we join the straight lines *EF*, *EK*, the prism in which the parallelogram *EBFG* is the base and the straight line *HK* its opposite is greater than the pyramid of which the triangle *EBF* is the base and the point *K* the vertex.

But the pyramid of which the triangle *EBF* is the base and the point *K* the vertex is equal to the pyramid of which the triangle *AEG* is the base and the point *H* the vertex; for they are contained by equal and similar planes.

Hence also the prism in which the parallelogram *EBFG* is the base and the straight line *HK* its opposite is greater than the pyramid of which the triangle *AEG* is the base and the point *H* the vertex.

But the prism in which the parallelogram *EBFG* is the base and the straight line *HK* its opposite is equal to the prism in which the triangle *GFC* is the base and the triangle *HKL* its opposite, and the pyramid of which the triangle *AEG* is the base and the point *H* the vertex is equal to the pyramid of which the triangle *HKL* is the base and the point *D* the vertex.

Therefore the said two prisms are greater than the said two pyramids of which the triangles *AEG*, *HKL* are the bases and the points *H*, *D* the vertices.

Therefore the whole pyramid, of which the triangle *ABC* is the base and the point *D* the vertex, has been divided into two pyramids equal to one another and into two equal prisms, and the two prisms are greater than the half of the whole pyramid. Q. E. D.