Let AB, PQ be commensurable, and let them contain their common measure m and ʼn times respectively. Through the points of division draw lines parallel to the sides of the parallelogram. Then the parallelograms will be divided into m and n equal parts respectively, (ii. 1. Cor. 2.)

and therefore

COR. I.

as their bases.

DABC :SPQR : m : n

:: AB: PQ.

Triangles of the same altitude are to one another

For a triangle is half the parallelograin on the same base and having the same altitude as the triangle.

THEOREM 5.

In the same circle or in equal circles angles at the centre and sectors are to one another as the arcs on which they stand.

Let ABC be a circle, of which O is the centre.

P

And let AOB, POQ be two angles at the centre.

Then AOB : POQ

arc AB arc PQ

:: sector AOB: sector POQ.

Let the angles AOB, POQ be commensurable, and let them contain their common measure m and n times respectively; and let the angles be divided into equal parts by radii.

Then (iii. 2) the areas and sectors are also divided into m and n equal parts respectively,

and therefore

arc AB arc PQ :: m : n

:: LAOB L POQ

:: sector AOB: sector POQ.

SECTION I.

SIMILAR FIGURES.

Def. 1. Similar rectilineal figures are those which have their angles equal, and the sides about the equal angles proportional.

Def. 2. Similar figures are said to be similarly described upon given straight lines, when those straight lines are homologous sides of the figures.

THEOREM I.

Rectilineal figures that are similar to the same rectilineal figure are similar to one another.

A

B

Proof. Let A, B be each of them similar to C; then will A be similar to B.

Proof. Since the angles of A and B are respectively

equal to the angles of C;

therefore also A and B are equiangular :

and since the sides about each angle of A are in the same ratio as the sides about the equal angle of C; and the sides about each of B are also in the same ratio,

therefore the sides about the equal angles of A and B are proportionals;

therefore A is similar to B.

(v. Def. 1.)

THEOREM 2.

If two triangles have their angles respectively equal, they are similar, and those sides which are opposite to the equal angles are homologous.

Let ABC, DEF be two triangles, which have the angles of the one equal to the angles of the other, viz. A, B, C respectively equal to D, E, F respectively;

Conceive the angle E placed on the angle B; then F and D would fall as F' and D' on BC and BA, or on those lines produced: and because the F= the C, therefore F'D' is parallel to CA;

Similarly by placing F on C, and D on A, the other proportions are obtained; and therefore the triangles are similar.

This theorem is a generalization of Theorem 15 in Book 1. If two angles and a side of one triangle are respectively equal to two angles and the corresponding side of another triangle, these triangles will be equal in all respects.

THEOREM 3.

If two triangles have one angle of the one equal to one angle of the other and the sides about these angles proportional, they are similar, and those angles which are opposite to the homologous sides are equal.

Let the triangles ABC, DEF have the angles at B and E equal, and let BA : BC :: ED : EF, then will the triangles be similar.

Conceive the angle E placed on the equal angle B, then D and F will fall as at D' and F' on the sides BA, BC,