PROPOSITION XXX. THEOREM.

The diagonals of a rhomboid bisect each other.

The diagonals AC, BD, of the rhomboid ABCD are mutually bisected in the point P.

For, since AB, CD, are parallel, the angles PAB, PBA, are respectively equal to the angles PCD, PDC (Prop. XIV.), and AB being also equal to CD, the triangles PAB, PCD, are equal (Prop.

A

B

XI.); therefore the sides AP, CP, opposite the equal angles ABP, CDP, are equal, as also the sides BP, DP, opposite the other equal angles. The diagonals of a rhomboid, therefore, bisect each other.

PROPOSITION XXXI. THEOREM. (Converse of Prop. XXX.) If the diagonals of a quadrilateral bisect each other, the figure is a rhomboid.

If the diagonals AC, BD (preceding diagram), bisect each other, ABCD is a rhomboid.

For the two sides AP, PB, and included angle being equal to the two sides CP, PD, and included angle, the side AB is equal to the side CD (Prop. VIII.). For similar reasons AD is equal to CB; hence (Prop. XXVIII.) the quadrilateral is a rhomboid.

BOOK II.

DEFINITIONS.

1. The altitude of a triangle is the distance of one of its sides, taken as a base, from the vertex of the opposite angle. The perpendicular AD from the vertex A

to the base BC, is the altitude of the triangle ABC.

2. The altitude of a rhomboid is the distance of one of its sides, considered as a base, from the opposite side.

3. The altitude of a trapezium is the distance between its parallel sides.

4. A rectangle is said to be contained by its adjacent sides.

The rectangle ABCD is contained by the sides DA, AB. For brevity it is often referred to as the rectangle of DA, AB.

5. If, within a rhomboid, two straight lines parallel to the adjacent sides be drawn so as to intersect the diagonal in the same point; then, of the four rhomboids into which the figure is divided, those two through which the diagonal passes are said to be about the diagonal, and the other two are called their complements.

In referring to a rhomboid it will be sufficient to employ the letters placed at two opposite corners.

PROPOSITION I. THEOREM.

The complements of the rhomboids about the diagonal of a rhomboid are equivalent.

Thus, in the above diagram, the rhomboids AG, GC, are equivalent.

The triangle ABD is equal to the triangle CDB; the triangle HGD to the triangle IDG, and the triangle EBG to the triangle FGB (Prop. XXVII. Cor. 3.); take the triangles HGD, EBG, from the triangle ABD, and there will remain the rhomboid AG; take, in like manner, from the other half of the rhomboid AC the triangles IDG, FGB, equal to the former two, and there will remain the rhomboid GC: these rhomboids, therefore, are equivalent.

PROPOSITION II. THEOREM.

Rhomboids are equal which have two sides, and the included angle in each equal.

Let the sides AB, AD, and the angle A in the rhomboid AC be respectively equal to the sides EF, EH, and the angle E in the rhomboid EG; these rhomboids are equal.

For the opposite sides

D

H

of rhomboids being equal,

it follows that the four

sides of the rhomboid AC

are respectively equal to

BE

those of the rhomboid EG; therefore, since the angles A and C are also equal, the two rhomboids are equal (Prop. XXV. Cor. B. I.)

Cor. 1. If a rhomboid and a triangle have two sides, and the included angle in the one respectively equal to two sides and

the included angle in the other, the rhomboid will be double the triangle (Prop. XXVII. Cor. 3. B. I.).

Cor. 2. Rectangles contained by equal lines are equal.

PROPOSITION III. THEOREM.

Rhomboids which have the same base and equal altitudes are equivalent.

Let the rhomboids AC, AE, standing upon the same base AB, have equal altitudes; or, which amounts to the same thing, let the opposite sides DC, FE, lie in the same line DE parallel to the base (Prop. XIII. Cor. 1. B. I.); these rhomboids are equal.

F

E

For DC is equal to FE, each being P C equal to AB (Prop. XXVII. B. I.); consequently DF is equal to CE and since DA, AF, are respectively equal to CB, BE, the triangle ADF is equal to the triangle BCE. Take the former triangle from the quadrilateral ABED, and there will remain the rhomboid AE; · take the latter triangle from the same space, and there will remain the rhomboid AC; these rhomboids are, therefore, equivalent.

B

Cor. 1. Rhomboids whose bases and altitudes are respectively equal are equivalent, for the equal bases being placed the one upon the other must coincide.

Cor. 2. Triangles whose bases and altitudes are respectively equal are equivalent, as they are the halves of equivalent rhomboids (Prop. XXVII. Cor. 3. B. I.).

Cor. 3. Every rhomboid is equivalent to a rectangle of equal base and altitude.

Cor. 4. A line bisecting the opposite sides of a rhomboid divides the rhomboid into two equal parts; and a line from the middle of any side of a triangle to the vertex of the opposite angle divides the triangle into two equal parts (Cor. 1 and 2.).

Cor. 5. Therefore a triangle is equivalent to a rhomboid of equal base and of half its altitude, or to one of equal altitude and of half its base.

Scholium.

1. It is very evident that the converse of the above proposi

tion is not true, that is to say, it cannot be inferred that two equivalent rhomboids shall have their bases and altitudes equal; for it has been shown (Prop. 1.) that the rhomboids AG, GC, are equivalent (see the diagram), where the base GF must be longer than the base GE, provided BA is longer than AD, for then the angle ADB being greater than ABD (Prop. XIX. B. I.), the angle EGB, which is equal to ADB, is greater than EBD; consequently EB is longer than EG, but EB is equal to GF, therefore GF is longer than GE.

2. It is however, true, that equivalent rhomboids upon the same base have equal altitudes, for if the altitude of one be supposed less than that of the other, and the side opposite its base be prolonged, a portion of the other rhomboid must be cut off thereby, and the remaining portion still be equal to the former rhomboid, by the proposition, which is absurd; the altitudes therefore are equal. Having shown this, we may further prove that equivalent rhomboids of equal altitudes have also equal bases, for they are equivalent to rectangles of the same bases and altitudes: now any side of a rectangle may be considered as the base; taking then those sides as bases which are equal to the altitude of the rhomboids, the other sides or altitudes are, as shown above, equal, and these altitudes are the bases of the rhomboids: the bases are therefore equal.

Cor. 6. Hence, equivalent triangles whose bases are equal, have equal altitudes; and equivalent triangles whose altitudes are equal, have equal bases (Cor. 5.).

If there be two straight lines of which one is divided into parts, the sum of the rectangles contained by the undivided line, and the several parts of the other, will be equal to the rectangle contained by the two whole lines.

Let the lines be AB, AC, of which the former is divided into the parts AD, DE, EB, then the rectangles contained by AC, and each of these parts, are together equal to the whole rectangle AH, contained by AB, AC.

Let DF, EG, be parallel to AC, then A D the angles FDE, GEB, being each equal to the angle A, the rhomboids AF, ÓG, EH, are rectangles, and DF, EG, being each equal to AC (Prop. XXVII. Cor.

IC

B

II