dently the double of DE; and consequently the difference between the squares of AE and CE, being equivalent to the rectangle contained by AC and the double of DE, is equivalent to twice the rectangle under AC and DE. Cor. The difference between the squares of the sides of a triangle, is equivalent to the difference between the squares of the segments of the base made by a perpendicular ;—a property likewise easily derived from the preceding proposition. PROP. XXII. THEOR. In any triangle, the sum of the squares of the sides, is equivalent to twice the square of half the base and twice the square of the straight line which joins the point of its bisection with the vertex. Let BD be drawn from the vertex B of the triangle ABC to bisect the base; the squares of the sides AB and BC are together equivalent to twice the squares of AD and DB. For let fall the perpendicular BE (I. 6.); and if the point D coincide with E, the triangle ABC being evidently isosceles, the squares of AB and BC are the same with twice the square of AB, or twice the squares of AE and EB, or of AD and DB (II. 10.) B DE But if the perpendicular fall upon C, the triangle is rightangled, and the squares of AB and BC are then equivalent to the square of AC, and twice the square of BC, or to twice the squares of AD, DC and BC; but (II. 10.) twice the squares of DC and BC are D equivalent to twice the square of DB, and consequently the squares of AB and BC are equivalent to twice the squares of AD and DB. B A DE C In every other case, whether the perpendicular BE fall within or without the base AC, the squares of AE, EC, the unequal segments of AC, are (II. 19. cor.) equivalent to twice the square of AD and twice the square of DE; add twice the square of EB to both, and the squares of AE, EB and of CE, EB-or the squares of the hypotenuses AB, BC-are equivalent to twice the square of AD, and twice the squares of DE, EB, that is, (II. 10.) to twice the square of DB. A. D CE PROP. XXIII. THEOR. The square of the side of a triangle is greater or less than the squares of the base and the other side, according as the opposite angle is obtuse or acute, by twice the rectangle contained by the base and the distance intercepted between the vertex of that angle and the perpendicular. In the oblique-angled triangle ABC, where the perpendicular BD falls without the base; the square of the side AB which subtends the oblique angle exceeds the squares of the sides AC and BC which contain it, by twice the rectangle under AC and CD. For the square of AD, or of the sum of AC and CD, is (II. 15.) equivalent to the squares of these lines AC, CD, together with twice their rectangle. Add the square of DB to each side, and the squares of AD, B C D valent to the square of AC, and the squares of CD, DB, together with twice the rectangle AC, CD; but the squares of CD, DB are (II. 10.) equivalent to the square of CB; whence the square of AB exceeds the squares of AC, BC, by twice the rectangle under AC and CD. Again, in the acute-angled triangle ABC, where the perpendicular BD falls within the triangle; the square of the side AB that subtends the acute angle, is less than the squares of the containing sides AC, BC, by twice the rectangle under the base AC and its intercepted portion CD. D C For the square of AD, or of the difference between AC and CD, is (II. 16.) equivalent to the squares of AC and CD, diminished by twice their rectangle. Add to each the square of DB, and the squares of AD and DB-or the square of AB--are equivalent to the square of AC, with the squares of CD and DB, or the square of BC, diminished by twice the rectangle under AC and CD. Consequently the square of AB is less than the squares of AC and BC, by twice the rectangle under AC and CD. Cor. If the triangle ABC be isosceles, having equal sides AC and BC, the square of the base AB is equivalent to twice the rectangle under the side AC, and the adjacent segment AD made by the perpendicular BD, whether the vertical angle be obtuse or acute. For the square of AB is equivalent to the squares of AC and BC, or twice the square of AC increased or diminished by twice the rect angle under AC and CD; that is, equivalent to twice the rectangle under AC and AD, the sum or difference of AC and CD. This might also be demonstrated from the corollary to Prop. 10. Schol. When the three sides of a triangle are given, the segments of the base made by a perpendicular may be found either by Prop. 21. or Prop. 23., and thence the perpendicular can easily be determined from the application of Prop. 10. But half the rectangle under this perpendicular and the base will, by corollary to Prop. 5., express the area of the triangle. PROP. XXIV. THEOR. The squares of the sides of a rhomboid, are together equivalent to the squares of its diagonals. Let ABCD be a rhomboid: The squares of all the sides AB, BC, CD, and AD, are together equivalent to the squares of the diagonals AC, BD. B For the angles BCE and CBE are equal to the alternate angles DAE and ADE, and the interjacent sides BC and AD are equal; wherefore (I. 20.) the triangles BEC and DEA are equal. Consequently CE being equal to E EA, the squares of AB, BC, are (II. 22.) equivalent to twice the square of AE and twice the square of BE; whence twice the squares of AB, BC, or the squares of all the sides of the rhomboid, are equivalent to four times the square of AE and four times the square of BE, that is, to the squares of AC and BD. |