Sol.—JoinBD. Construct a parallelogramEG[xlii.] equal to the triangleABD, and having the angleEequal to the given angleX; and to the right lineGHapply the parallelogramHIequal to the triangleBCD, and having the angleGHKequal toX[xliv.], and so on for additional triangles if there be any.ThenEIis aparallelogram fulfilling the required conditions.
Dem.—Because the anglesGHK,FEHare each equal toX(const.), they are equal to one another: to each add the angleGHE, and we have the sum of the anglesGHK,GHEequal to the sum of the anglesFEH,GHE; but sinceHGis parallel toEF, andEHintersects them, the sum ofFEH,GHEis two right angles [xxix.]. Hence the sum ofGHK,GHEis two right angles; thereforeEH,HKare in the same right line [xiv.].
Again, becauseGHintersects the parallelsFG,EK, the alternate anglesFGH,GHKare equal [xxix.]: to each add the angleHGI, and we have the sum of the anglesFGH,HGIequal to the sum of the anglesGHK,HGI; but sinceGIis parallel toHK, andGHintersects them, the sum of the anglesGHK,HGIis equal to two right angles [xxix.]. Hence the sum of the anglesFGH,HGIis two right angles; thereforeFGandGIare in the same right line [xiv.].
Again, becauseEGandHIare parallelograms,EFandKIare each parallel toGH; hence [xxx.]EFis parallel toKI, and the opposite sidesEKandFIare parallel; thereforeEIis a parallelogram; and because the parallelogramEG(const.) is equal to the triangleABD, andHIto the triangleBCD, the whole parallelogramEIis equal to the rectilineal figureABCD, and it has the angleEequal to the given angleX.HenceEIis a parallelogram fulfilling the requiredconditions.
It would simplify Problemsxliv.,xlv., if they were stated as the constructing of rectangles, andin this special form they would be better understood by the student, since rectangles are thesimplest areas to which others are referred.
Exercises.
Exercises.
1.Construct a rectangle equal to the sum of two or any number of rectilineal figures.
2.Construct a rectangle equal to the difference of two given figures.
PROP.XLVI.—Problem.On a given right line(AB)to describe a square.
PROP.XLVI.—Problem.On a given right line(AB)to describe a square.
Sol.—ErectADat right angles toAB[xi.], and make it equal toAB[iii.]. ThroughDdrawDCparallel toAB[xxxi.], and throughBdrawBCparallel toAD;thenACis the square required.
Dem.—BecauseACis a parallelogram,ABis equal toCD[xxxiv.]; butABis equal toAD(const.); thereforeADis equal toCD, andADis equal toBC[xxxiv.]. Hence the four sides are equal; thereforeACis a lozenge, and the angleAis a right angle.ThereforeACis a square(Def.xxx.).
Exercises.
Exercises.
1.The squares on equal lines are equal; and, conversely, the sides of equal squares areequal.
2.The parallelograms about the diagonal of a square are squares.
3.If on the four sides of a square, or on the sides produced, points be taken equidistant fromthe four angles, they will be the angular points of another square, and similarly for a regularpentagon, hexagon, &c.
4.Divide a given square into five equal parts; namely, four right-angled triangles, and asquare.
PROP.XLVII.—Theorem.In a right-angled triangle(ABC)the square on the hypotenuse(AB)is equalto the sum of the squares on the other two sides(AC,BC).
PROP.XLVII.—Theorem.In a right-angled triangle(ABC)the square on the hypotenuse(AB)is equalto the sum of the squares on the other two sides(AC,BC).
Dem.—On the sidesAB,BC,CAdescribe squares [xlvi.]. DrawCLparallel toAG. JoinCG,BK. Then because the angleACBis right (hyp.), andACHis right, being the angle of a square, the sum of the anglesACB,ACHis two right angles; thereforeBC,CHare in the same right line [xiv.]. In like mannerAC,CDare in the same right line. Again, becauseBAGis the angle of a square it is a right angle: in like mannerCAKis a right angle. HenceBAGis equal toCAK: to each addBAC, and we get the angleCAGequal toKAB. Again, sinceBGandCKare squares,BAis equal toAG, andCAtoAK. Hence the two trianglesCAG,KABhave the sidesCA,AGin one respectively equal to the sidesKA,ABin the other, and the contained anglesCAG,KABalso equal. Therefore [iv.] the triangles are equal; but the parallelogramALis double of the triangleCAG[xli.], because they are on the same baseAG, and between the same parallelsAGandCL. In like manner the parallelogramAHis double of the triangleKAB, because they are on the same baseAK, and between the same parallelsAKandBH; and since doubles of equal things are equal (Axiomvi.), the parallelogramALis equal toAH. In like manner it can be proved that the parallelogramBLis equal toBD.Hence the whole squareAFis equal to the sum of the two squaresAHandBD.
Or thus:Let all the squares be made in reversed directions. JoinCG,BK, and throughCdrawOLparallel toAG. Now, taking the∠BACfrom the right∠sBAG,CAK, the remaining∠sCAG,BAKare equal. Hence the△sCAG,BAKhave the sideCA=AK, andAG=AB, and the∠CAG=BAK; therefore [iv.] they are equal; and since [xli.] thePICTsAL,AHarerespectively the doubles of these triangles, they are equal. In like manner thePICTsBL,BDare equal; hence the whole squareAFis equal to the sum of the two squaresAH,BD.
This proof is shorter than the usual one, since it is not necessary to prove thatAC,CDare inone right line. In a similar way the Proposition may be proved by taking any of the eight figuresformed by turning the squares in all possible directions. Another simplification of the proof wouldbe got by considering that the pointAis such that one of the△sCAG,BAKcan beturned round it in its own plane until it coincides with the other; and hence that they arecongruent.
Exercises.
Exercises.
1.The square onACis equal to the rectangleAB.AO, and the square onBC=AB.BO.
2.The square onCO=AO.OB.
3.AC2−BC2=AO2−BO2.
4.Find a line whose square shall be equal to the sum of two given squares.
5.Given the base of a triangle and the difference of the squares of its sides, the locus of itsvertex is a right line perpendicular to the base.
6.The transverse linesBK,CGare perpendicular to each other.
7.IfEGbe joined, its square is equal toAC2+ 4BC2.
8.The square described on the sum of the sides of a right-angled triangle exceeds the square onthe hypotenuse by four times the area of the triangle (seefig.,xlvi., Ex.3). More generally, if thevertical angle of a triangle be equal to the angle of a regular polygon ofnsides, then theregular polygon ofnsides, described on a line equal to the sum of its sides, exceeds thearea of the regular polygon ofnsides described on the base byntimes the area of thetriangle.
9.IfACandBKintersect inP, and throughPa line be drawn parallel toBC, meetingABinQ; thenCPis equal toPQ.
10.Each of the trianglesAGKandBEF, formed by joining adjacent corners of the squares, isequal to the right-angled triangleABC.
11.Find a line whose square shall be equal to the difference of the squares on twolines.
12.The square on the difference of the sidesAC,CBis less than the square on the hypotenuseby four times the area of the triangle.
13.IfAEbe joined, the linesAE,BK,CL, are concurrent.
14.In an equilateral triangle, three times the square on any side is equal to four times thesquare on the perpendicular to it from the opposite vertex.
15.OnBE, a part of the sideBCof a squareABCD, is described the squareBEFG, havingits sideBGin the continuation ofAB; it is required to divide the figureAGFECDinto three partswhich will form a square.
16.Four times the sum of the squares on the medians which bisect the sides of a right-angledtriangle is equal to five times the square on the hypotenuse.
17.If perpendiculars be let fall on the sides of a polygon from any point, dividing each side intotwo segments, the sum of the squares on one set of alternate segments is equal to the sum of thesquares on the remaining set.
18.The sum of the squares on lines drawn from any point to one pair of opposite angles of arectangle is equal to the sum of the squares on the lines from the same point to the remainingpair.
19.Divide the hypotenuse of a right-angled triangle into two parts, such that the differencebetween their squares shall be equal to the square on one of the sides.
20.From the extremities of the base of a triangle perpendiculars are let fall on the oppositesides; prove that the sum of the rectangles contained by the sides and their lower segments is equalto the square on the base.
PROP.XLVIII.—Theorem.
PROP.XLVIII.—Theorem.
If the square on one side(AB)of a triangle be equal to the sum of the squares onthe remaining sides(AC,CB), the angle(C)opposite to that side is a rightangle.
Dem.—ErectCDat right angles toCB[xi.], and makeCDequal toCA[iii.]. JoinBD. Then becauseACis equal toCD, the square onACis equal to the square onCD: to each add the square onCB, and we have the sum of the squares onAC,CBequal to the sum of the squares onCD,CB; but the sum of the squares onAC,CBis equal to the square onAB(hyp.), and the sum of the squares onCD,CBis equal to the square onBD[xlvii.]. Therefore the square onABis equal to the square onBD. HenceABis equal toBD[xlvi., Ex. 1]. Again, becauseACis equal toCD(const.), andCBcommon to the two trianglesACB,DCB, and the baseABequal to the baseDB, the angleACBis equal to the angleDCB; but the angleDCBis a right angle (const.).Hence the angleACBis a rightangle.
The foregoing proof forms an exception to Euclid’s demonstrations of converse propositions, forit is direct. The following is an indirect proof:—IfCBbe not at right angles toAC, letCDbeperpendicular to it. MakeCD=CB. JoinAD. Then, as before, it can be proved thatADis equaltoAB, andCDis equal toCB(const.). This is contrary to Prop.vii.Hence the angleACBis aright angle.
Questions for Examination on Book I.
Questions for Examination on Book I.
1.What is Geometry?
2.What is geometric magnitude?Ans.That which has extension in space.
3.Name the primary concepts of geometry.Ans.Points, lines, surfaces, and solids.
4.How may lines be divided?Ans.Into straight and curved.
5.How is a straight line generated?Ans.By the motion of a point which has the same directionthroughout.
6.How is a curved line generated?Ans.By the motion of a point which continually changes itsdirection.
7.How may surfaces be divided?Ans.Into planes and curved surfaces.
8.How may a plane surface be generated.Ans.By the motion of a right line which crossesanother right line, and moves along it without changing its direction.
9.Why has a point no dimensions?
10.Why has a line neither breadth nor thickness?
11.How many dimensions has a surface?
12.What is Plane Geometry?
13.What portion of plane geometry forms the subject of the “First Six Books of Euclid’sElements”?Ans.The geometry of thepoint,line, andcircle.
14.What is the subject-matter of Book I.?
15.How many conditions are necessary to fix the position of a point in a plane?Ans.Two; forit must be the intersection of two lines, straight or curved.
16.Give examples taken from Book I.
17.In order to construct a line, how many conditions must be given?Ans.Two; as, forinstance, two points through which it must pass; or one point through which it must pass and a lineto which it must be parallel or perpendicular, &c.
18.What problems on the drawing of lines occur in Book I.?Ans.ii.,ix.,xi.,xii.,xxiii.,xxxi.,in each of which, except Problem 2, there are two conditions. The direction in Problem 2 isindeterminate.
19.How many conditions are required in order to describe a circle?Ans.Three; as, for instance,the position of the centre (which depends on two conditions) and the length of the radius (comparePost.iii.).
20.How is a proposition proved indirectly?Ans.By proving that its contradictory isfalse.
21.What is meant by the obverse of a proposition?
22.What propositions in Book I. are the obverse respectively of Propositionsiv.,v.,vi.,xxvii.?
23.What proposition is an instance of therule of identity?
24.What are congruent figures?
25.What other name is applied to them?Ans.They are said to be identically equal.
26.Mention all the instances of equality which are not congruence that occur in BookI.
27.What is the difference between the symbols denoting congruence and identity?
28.Classify the properties of triangles and parallelograms proved in Book I.
29.What proposition is the converse of Prop.xxvi., Part I.?
30.Defineadjacent, exterior, interior, alternateangles respectively.
31.What is meant by the projection of one line on another?
32.What are meant by the medians of a triangle?
33.What is meant by the third diagonal of a quadrilateral?
34.Mention some propositions in Book I. which are particular cases of more general ones thatfollow.
35.What is the sum of all the exterior angles of any rectilineal figure equal to?
36.How many conditions must be given in order to construct a triangle?Ans.Three; such asthe three sides, or two sides and an angle, &c.
Exercises on Book I.
Exercises on Book I.
1.Any triangle is equal to the fourth part of that which is formed by drawing through eachvertex a line parallel to its opposite side.
2.The three perpendiculars of the first triangle in question 1 are the perpendiculars at themiddle points of the sides of the second triangle.
3.Through a given point draw a line so that the portion intercepted by the legs of a given anglemay be bisected in the point.
4.The three medians of a triangle are concurrent.
5.The medians of a triangle divide each other in the ratio of 2 : 1.
6.Construct a triangle, being given two sides and the median of the third side.
7.In every triangle the sum of the medians is less than the perimeter, and greater thanthree-fourths of the perimeter.
8.Construct a triangle, being given a side and the two medians of the remainingsides.
9.Construct a triangle, being given the three medians.
10.The angle included between the perpendicular from the vertical angle of a triangle on thebase, and the bisector of the vertical angle, is equal to half the difference of the baseangles.
11.Find in two parallels two points which shall be equidistant from a given point, and whoseline of connexion shall be parallel to a given line.
12.Construct a parallelogram, being given two diagonals and a side.
13.The smallest median of a triangle corresponds to the greatest side.
14.Find in two parallels two points subtending a right angle at a given point and equallydistant from it.
15.The sum of the distances of any point in the base of an isosceles triangle from theequal sides is equal to the distance of either extremity of the base from the oppositeside.
16.The three perpendiculars at the middle points of the sides of a triangle are concurrent.Hence prove that perpendiculars from the vertices on the opposite sides are concurrent [seeEx.2].
17.Inscribe a lozenge in a triangle having for an angle one angle of the triangle.
18.Inscribe a square in a triangle having its base on a side of the triangle.
19.Find the locus of a point, the sum or the difference of whose distance from two fixed lines isequal to a given length.
20.The sum of the perpendiculars from any point in the interior of an equilateral triangle isequal to the perpendicular from any vertex on the opposite side.
21.The distance of the foot of the perpendicular from either extremity of the base of a triangleon the bisector of the vertical angle, from the middle point of the base, is equal to half the differenceof the sides.
22.In the same case, if the bisector of the external vertical angle be taken, the distance will beequal to half the sum of the sides.
23.Find a point in one of the sides of a triangle such that the sum of the intercepts made by theother sides, on parallels drawn from the same point to these sides, may be equal to a givenlength.
24.If two angles have their legs respectively parallel, their bisectors are either parallel orperpendicular.
25.If lines be drawn from the extremities of the base of a triangle to the feet of perpendicularslet fall from the same points on either bisector of the vertical angle, these lines meet on the otherbisector of the vertical angle.
26.The perpendiculars of a triangle are the bisectors of the angles of the triangle whose verticesare the feet of these perpendiculars.
27.Inscribe in a given triangle a parallelogram whose diagonals shall intersect in a givenpoint.
28.Construct a quadrilateral, the four sides being given in magnitude, and the middle points oftwo opposite sides being given in position.
29.The bases of two or more triangles having a common vertex are given, both in magnitudeand position, and the sum of the areas is given; prove that the locus of the vertex is a rightline.
30.If the sum of the perpendiculars let fall from a given point on the sides of a given rectilinealfigure be given, the locus of the point is a right line.
31.ABCis an isosceles triangle whose equal sides areAB,AC;B′C′is any secant cutting theequal sides inB′,C′, so thatAB′+AC′=AB+AC: prove thatB′C′is greater thanBC.
32.A,Bare two given points, andPis a point in a given lineL; prove that the difference ofAPandPBis a maximum whenLbisects the angleAPB; and that their sum is a minimum if itbisects the supplement.
33.Bisect a quadrilateral by a right line drawn from one of its angular points.
34.ADandBCare two parallel lines cut obliquely byAB, and perpendicularly byAC; andbetween these lines we drawBED, cuttingACinE, such thatED= 2AB; prove that the angleDBCis one-third ofABC.
35.IfObe the point of concurrence of the bisectors of the angles of the triangleABC, and ifAOproduced meetBCinD, and fromO,OEbe drawn perpendicular toBC; prove that the angleBODis equal to the angleCOE.
36.If the exterior angles of a triangle be bisected, the three external triangles formed on thesides of the original triangle are equiangular.
37.The angle made by the bisectors of two consecutive angles of a convex quadrilateral is equalto half the sum of the remaining angles; and the angle made by the bisectors of two opposite anglesis equal to half the difference of the two other angles.
38.If in the construction of the figure, Propositionxlvii.,EF,KGbe joined,
EF2+ KG2 = 5AB2.
39.Given the middle points of the sides of a convex polygon of an odd number of sides,construct the polygon.
40.Trisect a quadrilateral by lines drawn from one of its angles.
41.Given the base of a triangle in magnitude and position and the sum of the sides;prove that the perpendicular at either extremity of the base to the adjacent side, andthe external bisector of the vertical angle, meet on a given line perpendicular to thebase.
42.The bisectors of the angles of a convex quadrilateral form a quadrilateral whose oppositeangles are supplemental. If the first quadrilateral be a parallelogram, the second is a rectangle; if thefirst be a rectangle, the second is a square.
43.The middle points of the sidesAB,BC,CAof a triangle are respectivelyD,E,F;DGisdrawn parallel toBFto meetEF; prove that the sides of the triangleDCGare respectively equalto the three medians of the triangleABC.
44.Find the path of a billiard ball started from a given point which, after being reflected fromthe four sides of the table, will pass through another given point.
45.If two lines bisecting two angles of a triangle and terminated by the opposite sides be equal,the triangle is isosceles.
46.State and prove the Proposition corresponding to Exercise 41, when the base and differenceof the sides are given.
47.If a square be inscribed in a triangle, the rectangle under its side and the sum of the baseand altitude is equal to twice the area of the triangle.
48.IfAB,ACbe equal sides of an isosceles triangle, and ifBDbe a perpendicular onAC;prove thatBC2= 2AC.CD.
49.The sum of the equilateral triangles described on the legs of a right-angled triangle is equalto the equilateral triangle described on the hypotenuse.
50.Given the base of a triangle, the difference of the base angles, and the sum or difference ofthe sides; construct it.
51.Given the base of a triangle, the median that bisects the base, and the area; constructit.
52.If the diagonalsAC,BDof a quadrilateralABCDintersect inE, and be bisected in thepointsF,G, then
4△EF G = (AEB + ECD )− (AED + EBC ).
53.If squares be described on the sides of any triangle, the lines of connexion of the adjacentcorners are respectively—(1) the doubles of the medians of the triangle; (2) perpendicular tothem.