Let E H be parallel to B A. Then A H is to AC as BE is to BC [40]; but the intercepts D E and A H between the parallels E H and BA are equal to each other [I. 22]: therefore DE is to AC as BE is to B C, which W. T. B. D.

COROLLARY I. If two parallels subtend an angle, and if the portions which they determine on one side be equal, the subtending line nearer to the vertex is equal to half the other.

COROLLARY II. The intercepts of two parallels between the sides of vertical angles, are proportional to the distances from the vertex to their points of intersection with one side and its prolongation.

COROLLARY III. If two parallels subtend an angle and its vertical, and if the portion which they determine on one side and its prolongation, be equal, these lines are equal to each other.

COROLLARY IV. The intercepts of any number of parallels between. the sides of an angle, or vertical angles, are proportional to the distances from the vertex to their points of intersection with one side, or its prolongation.

COROLLARY V. The intercepts of parallels between the sides of an angle, or vertical angles, are proportional to their distances from the vertex.

THEOREM 42.

Two parallels are cut proportionally by three transversals drawn from the same point.

Let the transversals A B, A C, and A D, which cut the straight line BD in B, C and D, also cut its parallel EH in E, F and H.

Because the lines E F and BC are portions of two parallels intercepted between the sides of the angle BA C, the line E F is to BC as A F is to AC [41]; also, because FH and CD are portions of

the same parallels intercepted between the sides of the angle CAD,

the line F H is to CD as AF is to AC: therefore, E F is to BC as FH is to CD [i], and E F is to FH as BC is to CD [xiii], which W. T. B. D.

COROLLARY I. If two parallels be cut by transversals drawn from one point, and if the intercepts of one of them between the transversals, be equal to one another, the intercepts of the other are equal to one another.

COROLLARY II. Parallels are cut proportionally by any number of transversals drawn from the same point.

THEOREM 43.

Conversely, if the straight lines which pass through the homologous points of two parallels divided proportionally, meet cach other, they meet at the same point.

Let A B and CD be two parallels divided by the transversals A C, EF and BD, so that FD be to CF as E B is to A E.

H

D

F

C

E

Let H be the meeting point of AC and E F and let H B be joined, cutting CD in a point I. Because FI, CF, E B and A E are the intercepts of two parallels between three transversals from the same point, the line FI is to CF as E B is to AE [42]; but FD is to C F as E B is to AE [Hyp.]: therefore FI is equal to FD [xvii]; that is, the points D and I are one and the same point. Because the straight lines BD and B I have two points in common, they coincide throughout their extension [I. 4]; therefore BD meets A C and EF in the point H, which W. T. B. D.

Scholium. The straight lines passing through the extremities of two unequal straight lines having a common middle perpendicular, meet the latter in the same point (I. 30).

THEOREM 44.

(Eucl. VI. 3.)

The straight line subtending an angle is divided by the

bisectrix into portions, proportional to the distances from the

vertex to their intersections with the sides.

Let A C be a straight line subtending an angle ABC and let BD be the bisectrix of the angle, cutting A C in D.

Let the side B A of the angle be produced beyond the vertex, and let BH be the bisectrix of the adjacent angle CBE thus formed; let also CE be a

B.

H

parallel to DB through the point C: then the parallels DB and CE cut proportionally the sides of the angle CAE [38]; that is AD is to DC as AB is to BE. Because BH is perpendicular to BD [13 i], and E C is parallel to BD [Const.], the line BH is perpendicular to E C [I. 17], and E C subtends symmetrically the angle CBE [7 iii], that is, B E is equal

to BC: but AD

is to DC as AB is to BE [Dem.]; therefore A D is to DC as A B is to B C, which W. T. B. D.

COROLLARY I. Conversely, if the straight line which subtends an angle, be divided into portions proportional to the distances from the vertex to their intersections with the sides, the straight line joining the point of section to the vertex is the bisectrix of the angle.

COROLLARY II. If a straight line meet another, the prolongation of a straight line subtending one of the angle thus formed, to a point of the bisectrix of the adjacent angle, and the subtending line itself together with its prolongation, are proportional to the distances from the vertex to their intersections with the sides, and Conversely.

Scholium. The bisectrix of an angle bisects the straight line subtending the angle symmetrically (7).

SECTION II.

ON STRAIGHT LINES IN THE CIRCUMFERENCE.

DEFINITIONS.

21. Two straight lines are in medial ratio when the greater is a mean proportional between their sums and the lesser.

22. A straight line divided into two portions, is said to be medially divided, or divided in extreme and mean ratio, when the two portions thereof are in medial ratio: the greater portion is called the mean of the straight line, and the lesser its extreme.

THEOREM 45.

If there be a tangent and a secant to a circumference from a point without it, the tangent is a mean proportional to the secant and its outer part.

Let there be a tangent AB, and a secant A C of which the outer part is AD.

Let A E be the bisectrix of the angle BAC; let A B' be equal to