PROPOSITION I. THEOREM.

Triangles and parallelograms which have the same altitude, are to one another as their bases.

PART 1. Let the antecedent base be divided into any number of parts, and let one of those parts be repeated as often as possible on the consequent base; then connect the extremities of those parts with the common vertex.

Since all those small bases are = and the As on them of the same altitude, those ▲s are = (38. l. 1.), ..whatever submultiple one of those small bases is of the antecedent base, the same is one of the small as of the antecedent A, and as often as one of those small bases is contained in the consequent base, so often is one of the small As contained in the consequent A; and in like manner it can be shown that as often as any other submultiple of the antecedent base is contained in its consequent, so often is an equisubmultiple of the antecedent A contained in its consequent; ... the As are to one another as their bases, (Def. 5. 5.).

PART 2. Parallelograms of the same altitude, are to one another as their bases.

Let their diagonals be drawn ; then since the As which are the halves of these parallelograms (34. 1.), are of the same altitudes they are to one another as their bases (by part 1); . the parallelograms themselves are to one another as their bases.

Cor. 1. Triangles or parallelograms which have equal altitudes are to one another as their bases.

For the bases being placed in directum, the right line joining their vertices shall be parallel to the line in which their bases are; for the perpendiculars from their vertices on the bases are and par. Then it can be demonstrated as in the Prop., that those As are as their bases.

Cor. 2. Triangles and parallelograms on equal bases are to one another as their altitudes.

If the given As are right angled it is evident, if you consider the given bases as the altitudes, and vice versa. If they are not right angled, construct on their bases right angled As of the same altitudes with them; then those right angled As are as their altitudes; and the given As, which are to them (37. 1), are as their altitudes.

Note 1.-Hence it is evident that if the altitude of an isosceles ▲ be bisected, and right lines be drawn from the point of bisection to the extremities of the base, the whole A will be thus divided into four equal As.

2. If the right line from any of a ▲, bisecting the opposite side, be itself bisected; and if lines be drawn from this point of bisection to the extremities of the bisected side, the whole A will be thus divided into four equal as.

3. If an equilateral ▲ and a right angled isosceles ▲ be on the same base, they are to one another: /3: 1. For the O2 of the altitude of the equilat. 3 times the 2 of altitude of the right angled A... the altitudes are to one another 3: 1, but the As as their altitudes, .. &c.

4. If two such As have the same altitude they are to one another : : 1:3. For the of the base of the equilateral is of the of the base of the right angled 萎 A,.. those bases are to one another: : 1:3, and .. the As.

5. Therefore the equilateral A is a mean proportional between those two right angled As; and the two right angled As are to one another :: 1: 3.

PROP. II. THEOR.

If a right line be drawn parallel to any side of a triangle, it will cut the other two sides, or them produced into proportional segments: and the homologous segments are at the same side of the parallel line.

And if a right line cut two sides of a triangle, or those produced into proportional segments, so that the homologous segments be at the same side of it; it will be parallel to the remaining side.

PART 1. Connect the extremities of this line drawn par. with the opposite <s; then the As contained by the line drawn par., the connecting lines and the segments of the sides between the par. lines, are to one another, being on the same base (viz. the line drawn par.) and between the same parallels (viz. this line and the base of given A,). each of those As has the same ratio to the A contained by the line drawn par. and the segts. of the

sides between this line and the vertex of the given A, but one of those As: this latter :: the segt. (between par. lines) on which it stands the segt. in directum with it, (between line drawn par. and vertex of original A); and the other of those As: this same A:: the segt, between par lines on which it stands the segt. in directum with it, &c. as above, .. the first segt. in the first ratio has to the second segt. in first ratio the same ratio, which the first segt. in the second ratio has to the second segt. in the second ratio (18. 1. 5.) Therefore, &c.

:

PART 2. Those segts. being given in the ratio above mentioned, the line which thus cuts two sides of any A, or those produced, shall be parallel to the third side; for the ratio of those segts. being given the same, and they being to one another in the ratio of the two As before described to the same, those two As are equal, and being on the same base (i. e. the cutting line) they are (39. 1.) between the same parallels.

Cor. If several right lines be drawn parallel to the same side of a triangle, all the corresponding segments of the other sides shall be proportional.

From the extremity of the second segt. remote from the side par. draw a right line par. to the other cut side. Then those two segts. are as the segts. of this drawn line (by Prop.) but the segts, of this drawn line are to the corresponding segts. of the other side. Therefore, &c.

Note 1. The segments of any given line, in all cases mean those lines, which lie between the point of section and the extremities of the cut line, as in figures 71, plate 1 of Elrington, the segts. of the line AC cut by the perpendicular BD, are AÐ, DC, and either of the segts. may be greater than the cut line, i. e. when it produced is cut, as when the perpen. falls outside the A then the segt. DC is greater than the given line AC.

2. There are three variations of this Prop. i. e. where the sides are cut unproduced; where they are cut produced below the base, and where they are cut produced beyond the vertex. The segts... must be attended to; they are those lines which lie between the parallels, and those which lie between the cutting line and vertex of the given A. The As whose proportionality is treated of, must also be attended to; they are always as expressed in the above demonstration.

PROP. 111. THEOR.

A right line bisecting any angle of a triangle, divides the opposite side into segments proportional to the other two sides.

And if a right line drawn from any angle of a triangle, divide the opposite side into segments proportional to the other two sides, it bisects that angle.

PART 1. Draw through either extremity of the divided side a line parallel to the dividing line, and produce the side of the A not conterminous with this par. line to meet it (the parallel line).

Thes at the extremities of the line drawn parallel, and subtended by the produced part, and nonproduced uncut side, being each to the parts of the bisected (Prop. 29. 1) are to one another, and .. those lines subtending them are; but (Prop. 2. 6) the produced part the side of the given ▲ in continuum with it: the corresponding segments are to one another; .. the sides about the bisected are to one another, as the corresponding segments of cut side.

PART 2. The same construction being made; then since the segts. are to one another as the conterminous sides (by Hypoth.); and also since the segts. are to one another as one of those sides is to the produced part, (2. 1. 6) the other side is to the produced part; and.. the angles subtended by those lines are (5. 1); but in consequence of the parallels thoses are also respectively = to the parts into which the of the given ▲ is divided by the cutting line; .. those parts are, and.. the is L bisected.

Cor. Hence if a right line bisecting the vertical angle of a triangle also bisect the base, the triangle is isosceles.

Note 1. If the right line bisecting the external angle of a triangle, when produced meet the base, it will divide it into segments proportional to the other two sides.

The demonstration of this is precisely the same as that of the proposition, except that in the proposition the par. line may be drawn from either extremity of the cut side;

but in this case it must be drawn from that extremity adjacent to the point in which the bisecting line meets the produced part.

Note 2. The bisecting line will always meet the side produced, either on the side of the external or on the other side of it, except when the side about the vertical angle of an isosceles triangle is produced, the bisecting line will be then par. to the base.

It will meet it on the side of the external L when the greater side is produced, on the other side when the lesser is produced.

Note 3. If the internal and external angles at the vertex of a triangle be bisected, the base will be cut (by the bisectors), internally and externally in the same ratio.

PROP. IV. THEOR.

The sides about the equal angles of equiangular triangles are proportional; and those sides which subtend equal angles are homologous.

Let the As be so placed that two sides subtending =s may be conterminous, and in directum, that the As may lie towards the same parts, and that = <s may not be adjacent, but that thes which are adjacent may be each to its internal opposite < of the other A. Then on account of the respective equality of thoses, the sides subtending each pair will be parallel (28. 1); let the sides which are not conterminous be produced, and they will meet on the side on which the As lie (for the <s which they subtend, or the internal <s on the same side are, together less than two right <s (Ax. 12. 1).

Then in the whole A thus formed, there are parallels drawn to two of its sides (i. e. the sides of the given As, which are conterminous and not in directum, are par. to the sides produced to meet, which conterminous or par. sides are respectively to the produced parts); the segments of those sides made by the parallels are ... proportional (2. 6), but four of these segments are four sides of the given As (i. e. the sides placed in directum, and the sides produced to meet,) and the other two segts. (i. e. the produced parts,) are to their two remaining sides; and since those segments are proportional, the sides about