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Some Construction Problems Related to the Incircle of a Triangle

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Some Construction Problems Related to the Incircle of a Triangle SOME CONSTRUCTION PROBLEMS RELATED TO THE INCIRCLE OF A TRIANGLE by Amy B. Bell A Thesis Submitted to the Faculty of The Charles E Schmidt College of Science In Partial Fulfillment of the Requirements for the Degree of Masters of Science in Teaching Florida Atlantic University Boca Raton, Florida December 2006 Some Construction Problem s R elated to the Incircle of a Triangle by Amy B. B ell This thesis was prepared under the direction of the candidate's thesis advisor, Dr. Paul Yiu, Department of Mathematical Sciences, and has been approved by the members of her supervisory committee. It was submitted to the faculty of the Charles E. Schmidt College of Science and was accepted in partial fulfi llment of the requirements for the degree of Master of Science in Teaching of Mathematics. SUPERVISORY COMMITTEE: I I i.. • I (C-v Thesis Advisor Date ll Acknowledgements I would like to thank my graduate advisor, Professor Paul Yiu, for his support, patience and encouragement throughout the process of this thesis. I would also like to thank Professor Markus Schmidmeier for going beyond the usual duties of a thesis committee member and giving valuable feedback that '\,. made my thesis defense much more effective. I would also like to acknowledge the support of my mother, Katherine Gosney and my husband, Terry Bell. Both made it possible for me to spend the additional hours I needed to work through the mathematical ideas presented in this thesis. 111 ~-- ----- --------------- - Abstract Author: Amy B. Bell Title: Some Construction Problems Related to the Incircle of a Triangle Institution: Florida Atlantic University Thesis Advisor: Dr. Paul Yiu Degree: Masters of Science in Teaching Year: 2006 This thesis explores several construction problems related to the incircle of a triangle. Firstly, as a generalization of a theorem of D. W. Hansen, we find two quadruples of quantities related to a triangle which have equal sums and equal sums of squares. We also study the construction problems of triangles with centroid on the incircle, and those with a specified cevian - a median, an angle bisector, or an altitude- bisected by the incircle. Detailed analysis leads to dE>•:;i[.>' s of animation pictures using the dynamic software Geometer's Sketchpad. lV Contents List of Tables . v11 List of Figures . vm 1 Preliminaries 1 1.1 Introduction .... .. ... .. ............ .. 1.2 Notations ... ...... ............ ...... .... .. .. 3 1.3 The Heron formula and Heron triangles . 4 1.3.1 The Heron formula. 4 1.3 .2 Heron triangles . ... ....... ..... .. ... .. .. 4 1.4 Barycentric coordinates . 5 1.4.1 Homogeneous barycentric coordinates . 5 1.4.2 Absolute barycentric coordinates . 6 1.4.3 Equation ofthe incircle . ......... .. .. ....... 7 1.5 Geometric constructions . 7 1.5.1 Geometric solution of quadratic equations . 7 1.5.2 Non-constructibility .... .. .. ... .. .. ... .. .... 8 2 Hansen's theorem and generalizations 9 2.1 Hansen's theorem . 9 2.2 Integer relations from Pythagorean triangles . 11 2.3 Generalization of Hansen's theorem . 13 2.4 Some triangle geometry . 15 2.5 ProofofTheorem 2.3 .. .... ... .. ...... .... .. 18 2.6 Converse ofHansen's Theorem .. ........... .. ... 19 v 2. 7 Integer relations . 21 3 Triangles with centroid on the incircle 24 3.1 From s- bands-c. 25 3.2 From band c ........ .. ...... ........ .. ....... 28 3.3 Triangle with a median trisected by the incircle . ... ... 32 ';; 4 Triangle with midpoint of one cevian on incircle '•3 ' 4.1 Construction of triangle ABC with midpoint of A-median on incircle . 34 4.1.1 From s - b and s - c . 34 4.1 .2 From band c ........................... .. 36 4.2 Construction of triangle ABC with midpoint of A-bisector on incircle ............................ 39 4.2.1 From b and c . 40 4.2.2 An alternative construction . 42 4.2.3 From s- b and s - c . 43 4.3 Construction of triangle ABC with midpoint of A-altitude on incircle ............................ 44 4.3.1 From band c . .................... .. ...... 45 4.3 .2 From s- bands-c. 46 Bibliography VI ---- -- - - ---- -- - --- - List of Tables Integers with equal sums and sums of squares from Pythagorean triangles ... .... ...................... ....... 12 2 Integers with equal sums and sums of squares from Heron triangles . 21 3 More integers with equal sums and sums of squares from Heron triangles .......................... ............. 23 Vll ------- ------- - -- - ----- - · - List of Figures 1.1 The incircle . 3 1.2 Homogeneous barycentric coordinates . 6 1.3 Solutions of quadratic equations . 8 2.1 The incircle of a right triangle . 9 2.2 The incircle and excircles of a right triangle . 10 2.3 Circumcenter and orthocenter . 13 'i'" 2.4 Two quadruples with equal sums and equal sums of squares . 14 2.5 Incenter and excenter . 15 2.6 ] 'f a= 2R . ........ ...... ... ...... .. ......... ....... 16 2.7 ra + r b + rc = 4R + r. 17 3.1 Isosceles with centroid on incircle .. ..... .. ... ... ....... 25 3.2 Construction of u from v and w . 26 3.3 Triangle with centroid on incircle . 27 3.4 Triangle with centroid on incircle . 28 2 1 2 4 2 3.5 AP = -(b+ c) + -(b- c) . ... ... .. .... ... .. .. ........ 29 5 5 3.6 Solution of (3 .3) . 30 3.7 Triangles with centroid on incircle . 31 3.8 Triangle with a median trisected by the incircle .... .......... 32 4.1 Isosceles triangle with median bisected by incircle . 34 4.2 Triangle with median bisected by incircle . 35 4.3 Triangle with median bisected by incircle . 36 4.4 Solution of(4.1) ... ...... .. ............ .. .. ..... 37 vm -------- --- 4.5 Triangle with a median bisected by incircle . 38 4.6 Triangle with a median bisected by incircle . 38 4.7 Barycentric coordinates of midpoint of angle bisector . 39 4.8 Construction of A . .... ... .. ... .... .......... 40 2 4.9 Triangle with an angle bisector bisected by incircle . 42 4.10 Triangle with an angle bisector bisected by incircle . 43 4.11 Triangle with two altitudes bisected by incircle . 45 ~ .. 1 1 1 4.12 -+-=- . .. .. .... .. .. .... .. .... .. .. .. .. 46 p q r 1 1 1 4.13 -+-=- .. ... ...... .. ..... ... ... ... .. ...... 47 V W X 4.14 Construction of u from v and w . .. .. .... .. .. ... .... 48 4.15 Triangle with an altitude bisected by incircle . 48 IX Chapter 1 Preliminaries 1.1 Introduction In thi s thesis we study several problems re lated to the inc irc le of a triang le. In the present chapter we sha ll ex pl a in some bac kground results in triangle geometry th at sha ll be freely used in the subsequent chapters, namely, ( i) the Heron formula for the area of a triang le and formulas for various radi i asso­ c iated with a triang le (s1 .3), (ii) the noti on of barycentric coordinates and the descriptio n of the inc irc le in terms of such ( ~ 1.4), (iii ) a basic meth od o f solvin g quadratic equati ons as a ruler and compass construc­ ti on problem (§ 1.5). In Chapter 2, we generali ze a result of D. W. Hansen [I] and prove (Theorem 2.3) that for a given triang le with orthocentcr H , the two quadruples, namely, (i) the inradius and th e three exradii, and (ii ) the c ircumdiameter and the lengths of the segments AH, D H , C H , have equal sums and equal sums of squares. In §2.6, we prove a strong converse of Hansen's theorem (Theorem 2.4). In chapter 3, we study the constructi on problem of a triangle whose inc ircle contains the centro id . We analyze the problem w ith two sets of g iven data: ( i) g iven two le ngth s band c, to construct a third le ngth a so that they form a triang le w hose incirc le contains the centroid ( >3 3.2): (ii ) g iven two po ints /3, C: together w ith a third po in t X on the segment I3C, to construct a third po in t A suc h that ABC has its inc irclc containing the centroid, and tangent to BC at X, (§3. 1). A ft e r a deta il ed analysis, we incorporate the constructions with the use of the Geometer's Sketchpad, a dy namic software for geometric construc tions. F inally, in C hapter 4 , we study a variation of the the me of C hapter 3, na me ly the constructi on probl em o f a triang le w hose inc irclc b isects one specifi c cevian of the tri ang le. Recall that a ccvian is a line through a vertex of the tri ang le. T he cevians we consider in C hapte r 4 are a median, a n ang le bisector, and an a ltitude. T hus, we analyze each proble m below to the exte nt that it is solvable as a rule r and compass constructi on pro ble m . In each suc h case, we produce a constructi on using the Geometer's Sketc hpad . (i) Given two le ngth s band r, to construct a third le ngth a so that they form a triang le w hose incirc lc b isects (a) th e median (§4. 1.2), (b) the ang le bisector (§4.2. I), (c) the altitude Cfi4.3. 1) on the side a . (ii) Given two points B, C togethe r with a thi rd po in t X on the segment BC, to construct a triang le ABC w ith inc irc le tangent to BC at X and bisecting (a) th e median (§4. 1. 1), (b) the angle bisector (§4.2.3 ), (c) the altitude (§4.3.2) on the s ide BC. 2 - - - ------------- --------- ---. 1.2 Notations We consider a triangle ABC with side length s B C = a, C A = !J , AJJ = c.
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