Ex. 718. To construct a circle, touching two parallel lines and passing through a given point.

Ex. 719. About a given circle, to circumscribe a rhombus, having given an angle.

Ex. 720. In a given circle, to inscribe a rectangle, having given the ratio of two sides.

* Ex. 721. To divide a trapezoid into two similar trapezoids by a line parallel to the base.

Ex. 722. In the prolongation of the side AB of the triangle ABC to find a point X such that AX × BX = CX2.

Ex. 723. Through a given point P, to draw a line such that its distances from two other given points, R and S, shall have a given ratio.

* Ex. 724. Through a given point within a circle, to draw a chord so that its segments have a given ratio.

* Ex. 725. To construct a circle passing through a given point and touching two given lines.

* Ex. 726. Through a point of intersection of two circles, to draw a line forming chords which have a given ratio.

Ex. 727. To construct two lines, having given their mean proportional and their difference.

THEOREMS

Ex. 728. If a chord is bisected by another, either segment of the first is a mean proportional between the segments of the other.

Ex. 729. If in the triangle ABC the altitudes BD and AE meet in F, and AB BC, then

Ex. 730. Two triangles are similar if an angle of the one is equal to an angle of the other, and the altitudes corresponding with the other angles are proportional.

Ex. 731. If, between two parallel tangents, a third tangent is drawn, the radius is the mean proportional between the segments of the third tangent.

Ex. 732. If two circles are tangent externally, and through the point of contact a secant is drawn, the chords formed are proportional to the radii.

Ex. 733. If C is the midpoint of the arc AB, and a chord CD meets the chord AB in E, then

Ex. 734. If two circles intersect, their common chord produced bisects the common tangents.

Ex. 735. If an isosceles triangle is inscribed in a circle, the tangents drawn at the vertices form another isosceles triangle.

Ex. 736. The tangents drawn at the vertices of an inscribed rectangle enclose a rhombus.

Ex. 737. Two parallelograms are similar when they have an angle of the one equal to an angle of the other, and the including sides proportional.

Ex. 738. Two rectangles are similar if two adjacent sides are proportional.

Ex. 739. A circumference described upon the arm of an isosceles triangle as a diameter bisects the base.

Ex. 740. Three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians.

Ex. 741. If in the parallelogram

ABCD ZA = 60°, AC2 = AB2 + BC2 + AB × BC.

Ex. 742. If the altitude upon the hypotenuse of a right triangle divides the hypotenuse in extreme and mean ratio, the smaller arm is equal to the non-adjacent segment of the hypotenuse.

Ex. 743. If in a triangle the squares of two unequal sides have the same ratio as their projections, upon the third side, the triangle is a right triangle.

Ex. 744. If from a point 0, OA, OB, OC, and OD are drawn so that the angle AOB is equal to the angle BOC, and the angle BOD equal to a right angle, any line intersecting OA, OB, OC, and OD is divided harmonically. (287, 288.)

Ex. 745. The sum of the squares of the four sides of any quadrilateral is equal to the sum of the square diagonals plus four times the square of the line joining the midpoints of the diagonals. (326.)

* Ex. 746. If, from a point within the triangle ABC the perpendiculars OX, OY, and OZ be drawn upon AB, BC, and CA, respectively,

AX2 + BY2 + CZ2 = BX2 + YC2 + ZA2.

* Ex. 747. State and prove the converse of the preceding exercise. *Ex. 748. If two circles are tangent externally, the common tangent is a mean proportional between the diameters.

Ex. 749. If through a vertex C of triangle ABC a line CE is drawn parallel to AB, and any point H in CE is joined to F, the midpoint of AB, then FH is divided harmonically by BC and AC produced. (300.)

BOOK IV

AREAS OF POLYGONS

332. DEF. The unit of surface is a square whose side is the unit of length, e.g. a square inch, a square meter, etc.

333. The area of a surface is the number of units of surface it contains.

334. Two figures are equivalent if their areas are equal.

PROPOSITION I. THEOREM

335. Rectangles having equal altitudes are to each other as their bases.

D

m

A

B

Β'

Hyp. In rectangles ABCD and A'B'C'D', altitude AD= altitude A'D'.

Proof. CASE I. AB and A'B' are commensurable.

Let m be a common measure contained in AB 3 times, and in A'B' 4 times.

[To be completed by the student. Compare (280).]

'D

D'

FC'

A

B

CASE II. AB and A'B' are incommensurable.

E B

Divide AB into any number of equal parts, and lay off one

of these parts on A'B' as often as possible.

As AB and A'B' are incommensurable, there must be a remainder, EB', less than one of the equal parts.

By increasing the number of parts into which AB is divided, we can diminish the length of these parts, and, therefore, the length of EB' indefinitely.

Hence, A'E approaches A'B' as a limit, and A'EFD' approaches A'B'C'D' as a limit.

336. COR. Rectangles having equal bases are to each other as their altitudes.

337. NOTE. As a polygon can be measured by the unit of surface only, the words "triangle," "quadrilateral," etc., are frequently used for the area of a triangle, area of a quadrilateral, etc.

Ex. 750. From a given rectangle, to cut off another rectangle whose area is of the given one.

Ex. 751. To construct a rectangle, which is to a given one as m:n, when m and n are two given lines.

PROPOSITION II. THEOREM.

338. The areas of two rectangles are to each other as the products of their bases and altitudes.

Hyp. Rectangles R and R' have the bases b and b' and the altitudes a and a' respectively.

Proof. Construct the rectangle Q, having the same base as R', and the same altitude as R.

Ex. 752. Find the ratio of a rectangle 4 by 5 feet, and a square having a side of 10 feet.

Ex. 753. The diagonal of a rectangle is 26 inches long, and one of its sides 24 inches. The diagonal of another rectangle is 25 inches long, and one of its sides 20 inches. Find the ratio of the areas of the two rectangles.