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PROP. FFF. PROB.
GEN. ENUN.-To describe a square which shall be equal to the sum of any number of given squares.
PART. ENUN, and Const.-Let AB be the side of one of the gn. squares; from B draw BCI' to AB, and side of second gn. square ; join AC,
draw CD I' to AC, and
side of third gn. square ; join ad, and draw DE IN to AD, and = side of fourth gn. square ; join ae; then the square upon AE = sum of squares upon AB, BC, CD, DE.
DEMONST.-For since the / SADE, ACD, ABC are all rt. L' (Const.); .. (Prop. XLVII.) AE2 AD? + DE? AC? + CD + de ABS + BC? + CD + DEP.
And in the same manner, to any number of squares.Q. E. F.
Gen. Enun.--To describe a square which shall be equal to the difference of two squares, whose sides are given.
Part. ENUN, and Const.--Let A, B be the gn. sides of the 2 squares ; take a st. line cD terminated at c, but unlimited toward D; make ce A, and EF = B (Prop. III.); with cr. E, and rad. Ec, describe O CGD (Post. 3); 6 from F draw FG at rt. Ls to cd (Prop. XI.); then the square described upon rg is the square required.
Ep + GF (Prop. XLVII.); :: GF2 EG? EF
EF2 (Def. 15) = A? B? (Const.)
Hence a square may be described upon GF (Prop. XLVI.) = to the difference of the squares upon A and B. -Q. E. F.
GEN. ENUN.-If the square described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it, the angle contained by these two sides is a right angle.
PART. ENUN.—Let the square upon BC, one of the sides of the A ABC = the square upon the other two sides AB, AC; then 2 BAC is a rt. 2.
CONST.-From the pt. A draw AD at rt. Zs to AC (Prop. XI.); make AD = AB (Prop. III.), and join Dc.
DEMONST.–1. Because DA = AB, ... DA2 = AB2. To each of these add AC2; : . DA? + AC2 = BA? + AC2 (Ax. 2).
2. But, because DAC is a rt. 2, DA? + AC2 (Prop. XLVII.); and BC2 = BA2 + ACP (Hyp.), .. Dc2 = BC2, .. DC = BC.
3. And because DA = AB, and ac is common to A$ CAB, CAD, .. the two DA, AC = BA, AC, and the base Dc = base BC; .. the BAC = 2 DAC (Prop. VIII.) = rt. 2. (Const.)
Therefore, if the square &c.-Q. E. D.
In addition to the numerous Deductions which have been added to the several Propositions of Euclid, it may be advisable to subjoin a few to exercise the ingenuity of the Student. For this purpose, the Enunciation and Construction in some cases the Enunciation only—has been given; but the Problems involve no difficulty which, with a little consideration, he will not be able to surmount.
PROP. A. THEOR.--The sum of the diagonals of a trapezium is less than the sum of any four lines which can be drawn to the four angles from any point within the figure, except from the intersection of the diagonals.
(Deducible from Prop. XX.) PROP. ß. THEOR.—The sum of the sides of an isosceles triangle is less than the sum of the sides of any other tri. angle on the same base and between the same parallels.
(Deducible from Prop. XXVII. and Prop. P.) PROP. y. THEOR.–The difference of the angles at the base of any triangle is double the angle contained by a line drawn from the vertex perpendicular to the base, and another line bisecting the angle at the vertex.
Let ABC be a ^ ; draw AD l' to the base BC, and AE bisecting the 2 BAC; then L ABC Z ACB = 2 DAE.
(Deducible from Prop. XXXII.) PROP. O. THEOR.--If from one of the equal angles of an isosceles triangle a line be drawn to the opposite side, and from the same point a line be drawn to the opposite
side produced, so that the part intercepted may be equal to the former line, the angle contained by the side of the triangle and the first drawn line is double of the angle contained by the base and the latter.
Let ABC be an isosceles A; draw CD to any point in the opposite side AB; produce AB, aking DE
and join CE; then the L ACD
2 L BCE. E
(From Prop. XXXII.) PROP. E. THEOR.-If from one of the equal angles of an isosceles triangle a line, equal to one of the equal sides, be drawn to the opposite side, produced if necessary, the angle formed by this line and the base produced is equal to three times either of the equal angles of the triangle.
(From Prop. XXXII.) PROP. L. THEOR.-If the exterior angle of a triangle be bisected, and also one of the interior and opposite angles, the angle contained by the bisecting lines is equal to half the other interior and opposite angle.
(This Theorem, which will explain the principle of Hadley's Sextant, depends upon Prop. XXIX. Cor., and Prop. XXXII.)
PROP. 9. THEOR.--In any equilateral and equiangular polygon, of which the number of sides is even, the opposite sides are respectively parallel.
(From Prop. XXVIII. and XXXII.) PROP. 0.-If straight lines be drawn from the angles of a triangle, through any point within the triangle, to meet the sides, and the lines joining these points of intersection and the sides of the triangle be produced to meet, the three points of concourse will be in the same straight line.
Leta Bc be a A; from the angular pts. draw AD, BE, CF, through any pt. G within the A; join de, DF, EF, and
produce them to meet the sides in H, K, L; join HK, KL ; then HK, KL are in the same st. line.
(Deducible from Props. XIV, and XXXII.)
PROP. I. THEOR.-The diameters of a rhombus bisect each other at right angles.
PROP. K. PROB.--To inscribe a square in a gn. rhombus.
(By means of Prop. XXXII.)
PROP. N. PROB.-From one of the angles of a parallelogram to draw a line to the opposite side, which shall be equal to that side, together with the segment of it which is intercepted between the line and the opposite angle.
Let ABCD be the am; ACD the L from which the line is to be drawn; produce DB to E, and make BE = BD; join ce, and at the pt. c in the st. line EC make 6 ECF L CEF; then ch is the line required.
THEOR. — The triangle contained by the straight lines joining the points of bisection of the three sides of a given triangle, is one-fourth part of the given tri. angle, and is equiangular with it.
(Deducible from Prop. XLI.)
THEOR.-If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles opposite to either of the two equal sides be each a right angle, the triangles shall be equal and similar to each other.
(Deducible from Prop. XLVII.)
PROP. 0. THEOR.-If two exterior angles of a triangle be bisected, and from the point of intersection of the