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Draw BE. BD, GK, GI dividing the figs, into an= number of As, by lines from the equal Zs B and G.

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AB AABE ABCD ABDE Ax.l. . re=n

"FG AFGK AGHI - AGIK: Prop.68... ABP : FGR :: AABE + A BCD + A

BDE: AFGK + AGHI + AGIK; or AB : FGo :: whole fig. ABCDE : whole

fig. FGHIK. In like manner it may be proved that similar rectilinear figures of any number of sides, are as the squares of their like or corresponding sides.

Wherefore similar rectilinear figures, &c.

PROP. LXXXV. THEOR. The circumferences of equilateral polygons, which have the same number of sides, have the same ratio as the radii of their circumscribing circles.

Let ABCDEF, GHKILM be 2 equilat. polygons having an=no, sides inscrib. in Os whose centres are Q, R; join AQ, BQ, &c. and GR, HR, &c. : then shall Oce of polygon ABC, &c.:: * of polygon GHI &c. ::AQ : GR. AB

G H

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PROP. LXXXVI. THEOR. The areas of circles are to each other as the squares of their rudii, or as the squares of their diameters. (Figs. last prop.)

Prop. 83. For A AQF: A GRM :: AQS : GR ;

and equimultiples of As AQF: GRM have Prop. 70.

the same ratio ; .. Area of polygon ABC, &c. : area of poly

gon GHI, &c. :: AQP : GR”. If the sides be indefinitely diminished, and their number increased, the areas of the two polygons will be equal to the areas of their circumscribing Os; . . Area of O ABC, &c.: area of O GHI &c.

:: AQ: GR, Prop. 70.

or :: AD : GK”. Therefore the areas of circles, &c.

EXERCISES.

1. In a given straight line, to find a point equally distant from two given points.

2. Given one side of a right-angled triangle, and the difference between the hypothenuse and the other side; show how the triangle can be determined.

3. The straight line which bisects the angle opposite the base of an isosceles triangle, bisects the base also, and is at right angles to it.

4. In a given straight line to find a point from which, if lines be drawn to two given points on the same side of the line, these lines may make equal angles with it.

5. If the base of an isosceles triangle be bisected by a line drawn from the opposite angle, this angle will be also bisected, and the line will be perpendicular to the base.

6. The sum of the perpendiculars drawn from any point in one side of an equilateral triangle upon the other sides, is equal to the perpendicular from either of the angles, to its opposite side,

7. A perpendicular is the least distance of a given point from a given line.

8. The difference between any two sides of a triangle is less than the third side.

9. The difference between the sum of any two sides of a triangle and the third side, is less than twice the line drawn from any point in that side to the angle opposite.

10. If two straight lines bisect two sides of a triangle perpendicularly ; the perpendicular from the point of intersection of these lines upon the third side will bisect that side.

11. Through a given point to draw a straight line which shall make equal angles with two given straight lines.

12. A perpendicular to the base of an isosceles triangle to the opposite angle, will bisect that angle and the base.

13. If the base of a triangle be bisected by a straight line drawn from the opposite angle perpendicular to the base, that angle is bisected, and the triangle is isosceles.

14. If the angle of a triangle be bisected by a straight line perpendicular to the opposite side, that side is bisected, and the triangle is isosceles.

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