PROBLEMS & SOLUTIONSINS EUCLIDEAN

1 M. N. AREF AND WILLIAM WERNICKPROBLEMS &SOLUTIONSINSEUCLIDEANGEOMETRYDOVER BOOKS ON MATHEMATICSHANDBOOK OF MATHEMATICAL FUNCTIONS: WITH FORMULAS, GRAPHS, AND MATHEMATICALTABLES, Edited by Milton Abramowitz and Irene A. Stegun. (0-486-61272-4)ABSTRACT AND CONCRETE CATEGORIES: THE JOY OF CATS, Jiri Adamek, Horst Herrlich,George E. Strecker. (0-486-46934-4)NONSTANDARD METHODS IN STOCHASTIC ANALYSIS AND MATHEMATICAL PHYSICS, SergioAlbeverio, Jens Erik Fenstad, Raphael Hoegh-Krohn and Tom Lindstrom.(0-486-46899-2)MATHEMATICS: ITS CONTENT, METHODS AND MEANING, A. D. Aleksandrov, A. N. Kolmogo-rov, and M. A. Lavrent'ev. (0-486-40916-3)COLLEGE GEOMETRY: AN INTRODUCTION TO THE MODERN GEOMETRY OF THE TRIANGLE AND THECIRCLE,NathanAltshiller-Court. (0-486-45805-9)THE WORKS OF ARCHIMEDES, Archimedes. Translated by Sir Thomas Heath.(0-486-42084-1)REAL VARIABLES WITH BASIC METRIC SPACE TOPOLOGY, Robert B.

2 Ash. (0-486-47220-5)INTRODUCTION TO DIFFERENTIABLE MANIFOLDS, Louis Auslander and Robert E. MacKenzie.(0-486-47172-1)PROBLEM SOLVING THROUGH RECREATIONAL MATHEMATICS, Bonnie Averbach and OrinChein. (0-486-40917-1)THEORY OF LINEAR OPERATIONS, Stefan Banach. Translated by F. Jellett. (0-486-46983-2)VECTOR CALCULUS,Peter Baxandall and HansLiebeck. (0-486-46620-5)INTRODUCTION TO VECTORS AND TENSORS: SECOND EDITION-TWO VOLUMES BOUND AS ONE,Ray M. Bowen and C: C. Wang. (0-486-46914-X)ADVANCED TRIGONOMETRY, C. V. Durell and A. Robson. (0-486-43229-7)FOURIER ANALYSIS IN SEVERAL COMPLEX VARIABLES, Leon Ehrenpreis. (0-486-44975-0)THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 1, Euclid. Edited by Thomas L. Heath.(0-486-60088-2)THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 2, Euclid. (0-486-60089-0)THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 3, Euclid.

3 Edited by Thomas L. Heath.(0-486-60090-4)AN INTRODUCTION TO DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS, Stanley J. Farlow.(0-486-44595-X)PARTIAL DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS, Stanley J. Farlow.(0-486-67620-X)STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATIONS, Avner Friedman. (0-486-45359-6)ADVANCED CALCULUS, Avner Friedman. (0-486-45795-8)POINT SET TOPOLOGY, Steven A. Gaal. (0-486-47222-1)DISCOVERING MATHEMATICS: THE ART OF INVESTIGATION, A. Gardiner. (0-486-45299-9)LATTICE THEORY: FIRST CONCEPTS AND DISTRIBUTIVE LATTICES, George Gratzer.(0-486-47173-X)(continued on back flap)PROBLEMSANDSOLUTIONSINEUCLIDEAN GEOMETRYPROBLEMSANDSOLUTIONSINEUCLIDEAN GEOMETRYM. N. AREFA lexandria UniversityWILLIAM WERNICKCity College, NewYorkDOVER PUBLICATIONS, INCMINEOLA, NEW YORKC opyrightCopyright 1968 by Dover Publications, rights NoteThis Dover edition, first published in 2010, is a reissue of a work originallypublished by Dover Publications, Inc, in Standard Book NumberISBN-13: 978-0-486-47720-6 ISBN-10.

4 0-486-47720-7 Manufactured in the United States by Courier to my fatherwho has been always of lifelong inspirationand encouragement to EMPLOYED IN THIS BOOK xiiiCHAPTER 1 - Triangles and PolygonsTheorems and corollariesISolved problems4 Miscellaneous exercises25 CHAPTER 2 - Areas, Squares, and RectanglesTheorems and corollaries33 Solved problems35 Miscellaneous exercises56 CHAPTER 3 - Circles and TangencyTheorems and corollaries64 Solved problems67 Miscellaneous exercises95 CHAPTER 4 -Ratio and ProportionTheorems and corollaries105 Solved problemsio8 Miscellaneous exercises133 CHAPTER 5 Loci and TransversalsDefinitions and theorems144 Solved problems145 Miscellaneous exercises169 CHAPTER 6 -Geometry of Lines and RaysHARMONIC RANGES AND PENCILSD efinitions and propositions178 Solved problems182 ISOGONAL AND SYMMEDIAN LINES-BROCARD POINTSD efinitions and propositions186 Solved problems189 Miscellaneous exercises191ixXCONTENTSCHAPTER 7 -Geometry of the CircleSIMSON LINED efinitions and propositionsSolved problemsRADICAL AXIS-COAXAL CIRCLESD efinitions and propositionsSolved problemsPOLES AND POLARSD efinitions

5 And propositionsSolved problemsSIMILITUDE AND INVERSIOND efinitions and propositionsSolved problemsMiscellaneous exercisesCHAPTER 8 -Space GeometryTheorems and corollariesSolved problemsMiscellaneous exercisesINDEXPREFACEThis book is intended as a second course in EUCLIDEAN purpose is to give the reader facility in applying the theorems ofEuclid to the solution of geometrical PROBLEMS . Each chapter beginswith a brief account of Euclid's theorems and corollaries for simpli-city of reference, then states and proves a number of importantpropositions. Chapters close with a section of miscellaneous problemsof increasing complexity, selected from an immense mass of materialfor their usefulness and interest. Hints of solutions for a large numberof PROBLEMS are also this is not intended for the beginner in geometry, suchfamiliar concepts as point, line, ray, segment, angle, and polygonare used freely without explicit definition.

6 For the purpose of clarityrather than rigor the general term line is used to designate sometimesa ray, sometimes a segment, sometimes the length of a segment;the meaning intended will be clear from the of less familiar geometrical concepts, such as those ofmodern and space geometry, are added for clarity, and since theuse of symbols might prove an additional difficulty to some readers,geometrical notation is introduced gradually in each , 1968M. N. AREFWILLIAM WERNICKxiSYMBOLSEMPLOYED IN THIS BOOKL angleQcircleO parallelogramoquadrilateral0rectanglesqu are0trianglea = ba equals ba > ba is greater than ba < ba is less than baba is parallel to baba is perpendicular to ba / ba divided by ba : bthe ratio of a to bAB2the square of the distancefrom A to BbecausethereforexiiiCHAPTER 1 TRIANGLES AND POLYGONST heorems and CorollariesLINES AND If a straight line meets another straight line, the sum of the two adjacentangles is two right 1.

7 If any number of straight lines are drawn from a givenpoint, the sum of the consecutive angles so formed is, four right 2. If a straight line meets another straight line, the bisectorsof the two adjacent angles are at right angles to one If the sum of two adjacent angles is two right angles, their non-coincident arms are in the same straight If two straight lines intersect, the vertically opposite angles are If a straight line cuts two other straight lines so as to make the alternateangles equal, the two straight lines are If a straight line cuts two other straight lines so as to make: (i) twocorresponding angles equal; or (ii) the interior angles, on the same side of theline, supplementary, the two straight lines are If a straight line intersects two parallel straight lines, it makes: (i)alternate angles equal; (ii) corresponding angles equal.

8 (iii) two interiorangles on the same side of the line Two angles whose respective arms are either parallel orperpendicular to one another are either equal or Straight lines which are parallel to the same straight line are parallel toone AND THEIR If one side of a triangle is produced, (i) the exterior angle is equal to thesum of the interior non-adjacent angles; (ii) the sum of the three angles of atriangle is two right 1. If two angles of one triangle are respectively equal to twoangles of another triangle, the third angles are equal and the triangles 2. If one side of a triangle is produced, the exterior angle isgreater than either of the interior non-adjacent AND SOLUTIONS IN EUCLIDEAN GEOMETRYCOROLLARY 3. The sum of any two angles of a triangle is less than tworight If all the sides of a polygon of n sides are produced in order, the sum ofthe exterior angles is four right The sum of all the interior angles of a polygon of n sides is(2n - 4) right Two triangles are congruent if two sides and the included angle of onetriangle are respectively equal to two sides and the included angle of the Two triangles are congruent if two angles and a side of one triangleare respectively equal to two angles and the corresponding side of the If two sides of a triangle are equal, the angles opposite to these sides 1.

9 The bisector of the vertex angle of an isosceles triangle,(i) bisects the base; (ii) is perpendicular to the 2. An equilaterial triangle is also If two angles of a triangle are equal, the sides which subtend theseangles are An equiangular triangle is also Two triangles are congruent if the three sides of one triangle arerespectively equal to the three sides of the Two right-angled triangles are congruent if the hypotenuse and a sideof one are respectively equal to the hypotenuse and a side of the If two sides of a triangle are unequal, the greater side has the greaterangle opposite to If two angles of a triangle are unequal, the greater angle has the greaterside opposite to Any two sides of a triangle are together greater than the If two triangles have two sides of the one respectively equal to two sidesof the other and the included angles unequal.

10 Then the third side of that with thegreater angle is greater than the third side of the If two triangles have two sides of the one respectively equal to two sidesof the other, and the third sides unequal, then the angle contained by the sidesof that with the greater base is greater than the corresponding angle of Of all straight lines that can be drawn to a given straight line from agiven external point, (i) the perpendicular is least; (ii) straight lines whichmake equal angles with the perpendicular are equal; (iii) one making a greaterangle with the perpendicular is greater than one making a lesser AND POLYGONS3 COROLLARY. Two and only two straight lines can be drawn to a givenstraight line from a given external point, which are equal to one AND OVER FOUR-SIDED The opposite sides and angles of a parallelogram are equal, each diag-onal bisects the parallelogram, and the diagonals bisect one 1.