PROP. LXXV. THEOR. &. 6 En.

In a right angled triangle, if a perpendicu

lar be drawn from the right angle to the opposite side, the triangles on each side of the perpendicular are similar to the whole triangle, and to each other.

Let ABC be a rt. angled A, having the rt. Z BAC; draw ADI BC; then che As ABD, ADC are similar to the whole A ABC, and to one another.

Prop. 31. ... rem. LACB = rem. _ BAD;

And As ABC, ABD are equiangular; also the corresponding sides are proportionals (Prop. 73); and the A's are similar (Cor. Prop. 73).

In the same way it may be shown that AADC is equiangular and similar to ABC and also to A ABD.

Therefore in a rt, angled A, &c. Cor. The perpendicular AD is a mean proportional between the segments BD, DC, of the base BC. And each of the sides AB, AC, is a mean proportional between the base BC and the segment adjacent to the side.

For in the
Prop. 73. As, BDA, ADC, BD: DA :: DA: DC;

As, ABC, DBA, BC : BA :: BA : BD;
As ABC, ACD, BC : CA :: CA : CD.

PROP. LXXVI. THEOR. 33. 6 Eu. In the same or equal circles, angles at the cen

tres have the same ratio which the arcs on which they stand have to each other.

In the © ABD, whose centre is L, take any No. of = Ls DLK, KLI, ILH, &c.; then shall arc DC : arc CB :: LDLC : _CLB.

C. Are CB - ZCLB or Arc DC : Arc CB :: _DLC:Z CLB. Wherefore in the same, &c.

Since angles at the centre of a circle vary as the contained arcs upon which they stand ; it is frequently necessary for practical purposes to consider an arc as the measure of an angle; or the angle to be measured by the number of equal units or degrees contained in its corresponding arc, the whole circumference of the circle being usually divided into 360 degrees.

PROP. LXXVII. THEOR.
The angle made by the intersection of two

straight lines within a circle is measured by
half the sum of the two contained arcs.

Let str. lives BE and CD cut each other in pt. A, within the OBCFD; then will half the sum of the arcs DE and BC be the measure of the 4 of intersection BAC.

Draw EF || DC;

Join CE, Prop. 28.

IZDCE = ZCEF, Prop. 58. then .:: And = Ls at the Oce

stand on = arcs ;
.' Arc DE = arc CF.

To each add arc BC,
.. Arc DE + arc BC = Arc BCF.

But _ BEF at the Oce, or its equal Z BAC, is measured by half the arc BCF;

Arc BC + arc DE .:. _ BAC is measured by -

2 CoR.-Errors of " Excentricity” in instruments for the measurement of angles, are corrected by taking the half sum, or mean, between the readings of two opposite Verniers..

PROP. LXXVIII. THEOR.
If an angle of a triangle be bisected by a

straight line which cuts the base; the rect-
angle contained by the sides of the triangle
is equal to the rectangle contained by the
segments of the base, together with the
square of the bisecting line.

Let ABC be a A, and the str. line AD bisect the BAC; then shall BA. AC= BD. DC + ADP.

PROP. LXXIX. THEOR. If from any angle of a triangle a straight line be drawn perpendicular to the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle.