AD, BC, DC, diminished by twice the rectangle of BC, DC, that is (Prop. X.), to the squares of AC, BC, diminished by twice the rectangle of BC, DC.

In any triangle having an obtuse angle, the square of the side opposite thereto exceeds the squares of the base and other side, by twice the rectangle of the base and the distance of the perpendicular from the vertex of the obtuse angle.

In the triangle ABC, let B be an obtuse angle, AD the perpendicular on the prolongation of the base BC, then will the square of AC be equivalent to the squares of AB, BC, together with twice the rectangle of CB, BD.

For the square of AC is equivalent to the squares of CD, DA (Prop. X.), and the square of CD is equivalent to the squares of CB, BD, together with twice the rectangle of CB, BD (Prop. V.); therefore the square of AC is equivalent to the squares of the three lines CB, BD, DA, and twice the rectangle of CB, BD, that is, to the squares of CB, BA, and twice the rectangle of CB, BD.

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Cor. 1. From the two last propositions the converse of position X. immediately follows, that is, if the square of any side of a triangle be equivalent to the sum of the squares of the other two sides, the angle opposite the former shall be right; for these propositions show, that if such equivalence exist, the angle can neither be acute nor obtuse.

Cor. 2. We may, moreover, readily infer the converse of these two propositions themselves, that is, first, if in the triangle ABC (see the diagrams to Prop. XI.) the square of AB is equivalent to the squares of AC, BC, diminished by twice the rectangle of BC, CD, the angle C shall be acute; for by the above proposition and proposition X., if this angle were either obtuse or right, the said equivalence could not exist. Again, if in the triangle ABC the square of AC is equivalent to the squares of AB, BC, together with twice the rectangle of CB, BD, the angle B opposite AC shall be obtuse; for by last proposition, and proposition X., this angle can neither be acute nor right.

Scholium.

The last corollary may obviously be expressed in a more

unrestricted form, thus: If the square of any side of a triangle is less than the sum of the squares of the other two sides, the angle opposite the former side is acute, but if it is greater than that sum, the opposite angle is obtuse.

PROPOSITION XIII. THEOREM.

The squares of the sides of a rhomboid are together equivalent to the squares of the diagonals.

In the rhomboid ABCD, the squares of AB, BC, CD, DA, are together equivalent to the squares of AC, BD.

The truth of this for the rectangle has already been established (Prop. X. Cor. 4.). Let then the angles ABC, ADC, be obtuse, and consequently the other angles acute.

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Let BF be perpendicular to DC, and CE perpendicular to the production of AB.

Then, by last proposition, the square of AC is equivalent to the squares AB, BC, together with twice the rectangle of AB, BE; and (by Prop. XI.) the square of DB is equivalent to the squares of BC, DC, diminished by twice the equal rectangle DC, FC: for DC, FC, are respectively equal to AB, BE.

Hence, adding the squares of AC, BD, together, the sum is equivalent to the squares of AB, BC, CD, DA.

Cor. 1. Half the sum of the squares, that is, the squares of AB, BC, or of DC, CB, is equivalent to half the squares of BD, CA, that is, to twice the squares of BP, CP, (Prop. V. Cor.); hence, in any triangle, whether having an obtuse angle, as ABC, or having all its angles acute, as DBC, the sum of the squares of the two sides is equivalent to twice the squares of half the base, and of the line from the vertex to the middle of the base.

Cor. 2. Hence also, in any triangle, the squares on the sum and difference of the sides are equivalent to the squares of the base, and of twice the line from the vertex to the middle of the base. (Prop. VIII. Cor.).

PROPOSITION XIV. THEOREM. (Converse of Prop. XIII.) If the squares of the sides of a quadrilateral be together equivalent to the squares of the diagonals, the figure shall be a rhomboid.

In the quadrilateral ABCD, let the squares of the sides be equivalent to the squares of the diagonals, the figure is a rhomboid.

If it be not a rhomboid, the diagonals AC, BD, cannot bisect each other (Prop. XXXI. B. I.), let then m be the middle of AC, and n the middle of BD; join Dm, mB, and mn.

Then, by Cor. 1. last proposition, the squares of AD, DC, are together equivalent to twice the squares of Am, Dm; and the squares of AB, BC, are together equivalent to twice the squares of Cm, Bm; it there

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fore follows, that the squares of the four sides are equivalent to the squares of AC, and twice the squares of Dm, Bm. But, by hypothesis, the squares of the sides are equivalent to the squares of AC, BD; hence this latter square must be equal to twice the squares of Dm, Bm: but these are equivalent to the square of DB, together with four times the square of mn (Prop. XIII. Cor. I.); hence mn can have no value, that is, the middle of each diagonal must be one common point: therefore the figure is a rhomboid. (Prop. XXXI. B. I.)

Scholium.

The converse of the corollaries to proposition XIII. do not obtain. It will be sufficient to show this, with respect to the first corollary, the converse of which is as follows: If the sum of the squares of two sides of a triangle be equivalent to twice the square of a line, from the vertex to the base, together with twice the square of one of the parts, into which it divides the base; the base shall be divided in the middle.

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Let AD be the perpendicular from the vertex to the base of the triangle ABC, let DE be equal to BD, join AE, and let M be the middle of EC; then if AM be drawn, the squares of AC, AE, will be equivalent to twice the squares of AM, CM, but the square of AE is equal to the square of AB, since DE is equal to BD; therefore the squares of AC, AB, are equivalent to the squares of AM, CM, although M is not the middle of the base BC.

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BOOK III.

DEFINITIONS.

1. Every line which is not straight is called a curve line. 2. A circle is a space enclosed by a curve line, every point in which is equally distant from a point within the figure; which point is called the centre.

3. The boundary of a circle is called its circumference.

4. A radius is a line drawn from the centre to the circumference.

5. A diameter is a line which passes through the centre, and has its extremities in the circumference.

A diameter, therefore, is double the radius.

In the circle AEFBD, of which Cis the centre, CD is the radius, and AB the diameter.

6. An arc is any portion of the circumference.

7. The chord of an arc is the straight line joining its extremities. It is said to subtend the arc.

8. A segment of a circle is the portion included by an arc and its chord.

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The space EFGE included by the arc EFG, and the chord EG is a segment; so also is the space included by the same chord and the arc EADBG.

9. A sector of a circle is the portion included by two radii and the intercepted arc.

The space CBDC is a sector of the circle.

10. A tangent is a line which touches the circumference, that is, it has but one point in common with it, which point is called the point of contact.

11. One circle touches another when their circumferences have one point in common, and only one.

12. A line is inscribed in a circle when its extremities are in the circumference.

13. An angle is inscribed in a circle when its sides are inscribed.

14. A polygon is inscribed in a circle when its sides are inscribed, and under the same circumstances the circle is said to circumscribe the polygon.

Thus AB is an inscribed line, ABC an inscribed angle, and the figure ABCD is an inscribed quadrilateral.

15. A circle is inscribed in a polygon when its circumference touches each side, and the polygon is said to be circumscribed about the circle.

16. By an angle in a segment of a circle is to be understood, an angle whose vertex is in the arc, and whose

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sides intercept the chord; and by an angle at the centre is meant one whose vertex is at the centre. In both cases the angles are said to be subtended by the chords or arcs which their sides include.

POSTULATE.

From any point as a centre with any radius, a circumference may be described.

A diameter divides a circle and its circumference into two equal parts; and, conversely, the line which divides the circle into two equal parts is a diameter. Let AB be a diameter of the circle AEBD, then the portions AEB, ADB, are equal both in surface and boundary.

Suppose the portion AEB were to be applied to the portion ADB, while the line AB still remains common to both, there must be an entire coincidence; for if any part of the boundary AEB were to fall either within or without the boundary ADB, lines from the centre to the circumference could not all be equal. Therefore a diameter divides the circle and its circumference in two equal parts.

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Conversely, the line dividing the circle into two equal parts is a diameter.

For, let AB divide the circle into two equal parts, then, if the centre is not in AB, let AF be drawn through it, which is,