49. If three circles intersect each other, their common chords intersect in the same point. [§ 528.]

50. In any inscribed quadrilateral, the product of the diagonals is equal to the sum of the products of the opposite sides.

51. To inscribe a square in a given semicircle.

52. To inscribe a square in a given

triangle.

53. ABCD is a parallelogram, E a point on BC such that BE is one fourth of BC. AE cuts the diagonal BD in F. Show that BF is one fifth of BD.

54. Two chords of a circle drawn from a common point A on the circumference and cut by a line parallel to a tangent through A, are divided proportionally. [Suggestion. Join the extremities of the chords and prove the triangles similar.]

BOOK IV

561. DEFINITIONS. We measure a magnitude by comparing it with a similar magnitude that is taken as the unit of measure. If we wish to find the length of a line, we find how many times a linear unit of measure, say a foot, is contained in the line. This number, with the proper denomination, is called the length of the line.

Similarly, we measure any portion of a surface by comparing it with some unit of surface measure. We find how many times this unit, say a square yard, is contained in the portion of surface. This number, with the denomination square yards, we call the area or superficial content of the surface measured.

Polygons that have the same areas are equivalent polygons. Equivalent polygons are not necessarily equal in all respects. They need not even have the same number of sides. For example, a triangle, a square, and a hexagon may be equivalent. The base of a polygon is primarily the side upon which the figure stands; but usage has sanctioned a more extended application of the term. Any side of a polygon may be considered the base. In a parallelogram, if two opposite sides are horizontal lines, they are frequently called the upper and lower bases of the parallelogram. In a trapezoid, the two parallel sides are called its bases.

The altitude of a parallelogram is the perpendicular distance between two opposite sides. A parallelogram may therefore

have two different altitudes.

The altitude of a trapezoid is the perpendicular distance between its bases.

PROPOSITION I. THEOREM

562. Parallelograms having equal bases and equal altitudes are equivalent.

B

F

A

E

H

Let ABCD and EFGH be two parallelograms having equal bases and equal altitudes.

To Prove ABCD and EFGH equal in area.

Proof. Place EFGH upon ABCD so that their lower bases shall coincide. Because they have equal altitudes their upper bases are in the same line.

Prove AIB and DJC equal.

The parallelogram AIJD is composed of the quadrilateral ABJD and the ▲ AIB.

The parallelogram ABCD is composed of the quadrilateral ABJD and the ADJC.

563. EXERCISE. Rectangles having equal bases and altitudes are equal in all respects.

564. EXERCISE. Construct a rectangle equivalent to a given parallelogram.

565. EXERCISE. Prove Prop. I., using this figure:

566. EXERCISE. Construct a rectangle whose area is double that of a given equilateral triangle.

B

C E

F

567. EXERCISE. A line joining the middle points of two opposite sides of a parallelogram divides the figure into two equivalent parallelograms.

568. Triangles having equal bases and equal altitudes are equivalent.

Let the A ABC and DEF have equal bases and equal altitudes. To Prove the AABC and DEF equal in area.

Proof. On each triangle construct a parallelogram having for its base and altitude the base and altitude of the triangle. These parallelograms are equivalent. (?) .. the triangles are equivalent. (?)

Q.E.D.

569. COROLLARY I. If a triangle and a parallelogram have equal bases and equal altitudes, the triangle is equivalent to one half the parallelogram.

570. COROLLARY II. To construct a triangle equivalent to a given polygon.

To construct a triangle equivalent to ABC ... G.

Draw BD.

Through draw CX parallel to BD, meeting AB prolonged at X.

Draw DX.

Show that ABXD and BCD have a common base and equal altitudes. .. ABXD = polygon AXDEFG = polygon ABCDEFG.

E

ABCD, and the