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sequently the angles FBD and FBE are equal, and each of them the half of DBE. The angle FBD, being therefore one third part of a right angle, and the angle DBA two third parts, the whole angle FBC must be an entire right angle, or the straight line BF is perpendicular to AB.

PROP. XXXIX. PROB.

On a given finite straight line to construct a square.

Let AB be the side of the square which it is required to

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For, by this construction, the figure has all its sides equal, and one of its angles ABC a right angle; which comprehends the whole of the definition of a square.

PROP. XL. PROB.

To divide a given straight line into any number of equal parts.

Let it be required to divide the straight line AB into a given number of equal parts, suppose five.

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For draw DM, EN, FO, and GP parallel to AB. And because DH is parallel to EM, the exterior angle ADH is equal to DEM (I. 25.); and for the same reason, since AH is parallel to DM, the angle DAH is equal to EDM. Wherefore the triangles ADH and DEM, having two angles respectively equal, and the interjacent sides AD, DE, are (1. 23.) equal, and consequently AH is equal to DM. In the same manner, the triangle ADH is proved to be equal to EFN, FGO, and GCP, and therefore their bases EN, FO, and GP are all equal to AH. But these lines are equal to HI, IK, KL, and LB, for the opposite sides of parallelograms are equal (I. 29.). Wherefore the several segments AH, HI, IK, KL, and LB, into which the straight line AB is divided, are all equal to each other.

1

1 1

ELEMENTS

OF

GEOMETRY.

BOOK II.

DEFINITIONS.

1. In a right-angled triangle, the side that subtends the right angle is termed the hypotenuse; either of the sides which contain it, the base; and the other side, the perpen

dicular.

2. The altitude of a triangle is a perpendicular let fall from its vertex upon the extension of its base.

3. The altitude of a trapezoid is the perpendicular drawn from one of its parallel sides to the other.

4. The complements of rhomboids about the diagonal of a rhomboid, annexed to either of them, forms what is termed a gnomon.

5. A rhomboid or rectangle is said to be contained by any two adjacent sides.

PROP I. THEOR.

Triangles which have the same altitude, and stand on the same base, are equivalent.

The triangles ABC and ADC which stand on the same base AC and have the same altitude, contain equal spaces. For join the vertices B, D by a straight line, which produce both ways; and from A draw AE (I. 26.) parallel to CB, and from C draw CF parallel to AD.

Because the triangles ABC, ADC have the same altitude, the straight line EF is parallel to AC (I. 27.), and consequently the figures CE and

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is equal to BF; but EA is equal to

BC (1. 29.), and the interior angle AED is equal to the exterior angle CBF (I. 25.). Thus the two triangles EDA, BFC have the sides ED, EA equal to BF, BC, and the contained angle AED equal to CBF, and are, therefore, equal (I. 3.). Take these equal triangles CBF and EDA from the whole quadrilateral space AEFC, and there remains the rhomboid AEBC equal to ADFC. Whence the triangles ABC and ADC, which are the halves of these rhomboids, (I. 29. cor.) are likewise equal.

Cor. Hence the rhomboids on the same base and between the same parallels are equivalent.

PROP. II. THEOR.

Triangles which have the same altitude and stand on equal bases, are equivalent.

The triangles ABC, DEF, standing on equal bases AC

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