### Mathematics Teacher

#### Geometry Bibliography: Triangles

Hubert Ludwig, Ball State University

 ```Morgan's Theorem Tad Watanabe, Robert Hanson, and Frank D. Nowosielski Investigating the area of a hexagon formed in the interior of a triangle by certain n- sectors of the angle. (889, 1996) 420 - 423 Using a Surface Triangle to Explore Curvature James Casey Investigating triangle angle sums on various surfaces, e.g., bananas, soap bottles, watermelons, etc. (87, 1994) 69 - 77 The Sidesplitting Story of the Midpoint Polygon Y. David Gau and Lindsay A. Tartre The midline of a triangle theorem. Varignon's theorem. Extensions to pentagons and other polygons. (87, 1994) 249 - 256 Golden Triangles, Pentagons, and Pentagrams William A. Miller and Robert G. Clason Informal investigations of recursion. The golden ratio, Fibonnaci sequence, regular polygons, and pentagrams. (87, 1994) 338 - 344, 350 - 353 Counting Embedded Figures Timothy V. Craine Activities. How many triangles, squares, rectangles, etc., are there in a given figure? (87, 1994) 524 - 528, 538 - 541 Heron's Remarkable Triangle Area Formula Bernard M. Oliver Heron's proof and a modern short proof. (86, 1993) 161 - 163 The Use of Dot Paper in Geometry Lessons Ernest Woodward and Thomas Ray Hamel Area, perimeter, congruence, similarity, Cevians. (86, 1993) 558 - 561 SSA and the Steiner-Lehmus Theorem David Beran Conclusions which can be drawn from an SSA correspondence and a proof of the Steiner-Lehmus theorem. (85, 1992) 381 - 383 Area of a Triangle Donald W. Stover Sharing Teaching Ideas. An alternate method for finding the area of a triangle given the lengths of the sides. (83, 1990) 120 Why is the SSA Triangle Congruence Theorem Not Included in Textbooks? Daniel B. Hirschhorn A plea for the inclusion of a special instance of SSA. (83, 1990) 358 - 361 SSA: The Ambiguous Case Carolyn J. Case Sharing Teaching Ideas. Presents a chart for investigating SSA. (82, 1989) 109 - 111 Triangles of Equal Area and Perimeter and Inscribed Circles Jean E. Kilmer A triangle has equal area and perimeter if and only if it can be circumscribed about a circle of radius 2. (81, 1988) 65 - 70 Primitive Quadruples for the Law of Cosines Mark A. Mettler Investigating a*a = b*b + c*c - 2bc cos A for A = 60, 90, 120 degrees. (81, 1988) 306 - 308 Reopening the Equilateral Triangle Problem: What Happens If . . . Douglas L. Jones and Kenneth L. Shaw Investigations arising from a question about the sum of the distances from an interior point to the sides of an equilateral triangle. (81, 1988) 634 - 638 Let ABC Be Any Triangle Baruch Schwartz and Maxim Bruckheimer Drawing a triangle that does not look special. (81, 1988) 640 - 642 The Peelle Triangle Alan Lipp Information which can be deduced from the triangle about points, lines, segments, squares, and cubes. A relation to Pascal's triangle. 80, (1987) 56 - 60. Two Views of Oz John Pancari and John P. Pace Using the Scarecrow's Pythagorean-like utterance to define the fundamental isosceles triangle of Oz. 80, (1987) 100 - 101. Another Approach to the Ambiguous Case Bernard S. Levine Using the law of cosines to set up a quadratic equation. 80, (1987) 208 - 209. A Geometric Proof of the Sum-Product Identities for Trigonometric Functions Joscelyn Jarrett Utilizing points on a unit circle. 80, (1987) 240 - 244. Rethinking the Ambiguous Case Allen L. Peek Again relating the solution of the problem to the solution of a quadratic equation. 80, (1987) 372. Illustrating the Euler Line James M. Rubillo Finding the coordinates of the points on the line. 80, (1987) 389 - 393. Some Theorems Involving the Lengths of Segments in a Triangle Donald R. Byrkit and Timothy L. Dixon Proof of a theorem concerning the length of an internal angle bisector in a triangle. Other related results are included. 80, (1987) 576 - 579. Integer-sided Triangles and the SSA Ambiguity Abraham M. Glicksman Some results concerning integer-sided triangles that contain a 60-degree angle. 80, (1987) 580 - 584. Problem Solving in Geometry - a Sequence of Reuleaux Triangles James R. Smart Investigation of area relations for a sequence of Reuleaux triangles associated with an equilateral triangle and a sequence of medial triangles. 79, (1986) 11 - 14. Did the Scarecrow Really Get A Brain? Lowell Leake An analysis of the Scarecrow's Pythagoras-like statement in The Wizard of Oz. 79, (1986) 438 - 439. A Property of Right Triangles and Some Classical Relations Angelo S. DiDomenico A Pythagorean triple relation which leads to other results including Heron's formula and the law of cosines. 79, (1986) 640 - 643. Drawing Altitudes of Triangles Susan A. Brown Graph paper exercises to reinforce the definition. 78, (1985) 182 - 183. An "Ancient/Modern" Proof of Heron's Formula William Dunham Utilizing Heron's inscribed circle and some trigonometric results. 78, (1985) 258 - 259. Investigating Shapes, Formulas, and Properties With LOGO Daniel S. Yates Logo activities leading to results on areas and triangle geometry. 78, (1985) 355 - 360. (See correction p. 472.) Triangles, Rectangles, and Parallelograms Melfried Olsen and Judith Olsen Activities involving the manipulation of models of geometric figures. 76, (1983) 112 - 116. An Improvement on SSA Congruence for Geometry and Trigonometry Shraga Yushurum and David C. Kay Conditions under which SSA yields congruence. A result for non-Euclidean geometry is also presented. 76, (1983) 364 - 367. SSA: When Does It Yield Triangle Congruence? Bonnie H. Litwiller and David R. Duncan One of three further conditions will guarantee the result. 74, (1981) 106 - 108. Area = Perimeter Lee Markowitz When is the area of a triangle equal to its perimeter? 74, (1981) 222 - 223. Those Amazing Triangles Christian R. Hirsch Activities. Morley's triangle. Outer Napoleon triangle. 74, (1981) 444 - 448. 80 Proofs From Around The World Tony Trono Proofs of "If the angle bisectors of a triangle are equal then the triangle is isosceles." 74, (1981) 695 - 696. The Golden Mean and An Intriguing Congruence Problem David L. Pagini and Gerald E. Gannon Another approach to triangles which have five non-corresponding congruent parts but which are not congruent. 74, (1981) 725 - 728. Isosceles Triangles With The Same Perimeter and Area Mark E. Bradley There can never be more than two isosceles triangles having a given number as both perimeter and area. 73, (1980) 264 - 266. Area and Cost Per Unit: An Application Jan J. Vandever Activities for practice with area formulas. 73, (1980) 281 - 284, 287. Beyond The Usual Constructions Melfried Olson Activities leading to the Fermat point, Simpson line, etc. 73, (1980) 361 - 364. Exploring Congruent Triangles A. S. Green A suggested method for the introduction of congruence of triangles. 73, (1980) 434 - 436. Graphing - Perimeter - Area Merrill A. Meneley Activities dealing with areas and perimeters of polygons. Uses a coordinate system. 73, (1980) 441 - 444. Serendipity On The Area Of A Triangle Madelaine Bates When are the area and the perimeter of a triangle equal? 72, (1979) 273 - 275. Right Or Not: A Triangle Investigation Daniel T. Dolan Activities leading to the development of relations between the lengths of the sides of a triangle and its classification according to angle size. 72, (1979) 279 - 282. Pie Packing William Jacob Bechem Activities for investigating properties of 30 - 60 and isosceles right triangles. 72, (1979) 519 - 522. Tiling Richard A. Freitag Activities involving the replication of figures. Congruence and similarity. 71, (1978) 199 - 202. Computer Classification Of Triangles and Quadrilaterals - A Challenging Application J. Richard Dennis Computer application, uses coordinates of vertices. 71, (1978) 452 - 458. Tetrahexes Raymond E. Spaulding Activities involving congruence and symmetry. 71, (1978) 598 - 602. Problem Posing and Problem Solving: An Illustration Of Their Interdependence Marion A. Walter and Stephen I. Brown Given two equilateral triangles, find a third whose area is the sum of the areas of the first two. The Pythagorean Theorem and a generalization. 70, (1977) 4 - 13. The Key Duplicator: A Congruence Machine Caroline Hollingsworth The relation of a key duplicator to congruence of geometric figures. 70, (1977) 127 - 128. Some Novel Consequences Of The Midline Theorem Larry Hoehn Application of the theorem concerning the segment joining the midpoints of the sides of a triangle. 70, (1977) 250 - 251. Almost Congruent Triangles With Integral Sides John T.F. Briggs Triangles having five parts of one congruent to five (non-corresponding) parts of the other. 70, (1977) 253 - 257. An Investigation Of Integral 60 and 120 Triangles Richard C. Muller Law of cosines investigation. Computer related. 70, (1977) 315 - 318. Notes On The Partial Converses Of A Familiar Theorem William M. Waters, Jr. Partial converses of the result that for an isosceles triangle the bisector of the apex angle, the altitude to the base and the median to the base are identical. 70, (1977) 458 - 460. Congruence Extended: A Setting For Activity In Geometry Gail Spittler and Marian Weinstein Special (two part) triangle congruence theorems. Quadrilateral congruence theorems. 69, (976) 18 - 21. Some Additional Results Involving Congruence Of Triangles Norbert J. Kuenzi, John A. Oman, and Robert W. Prielipp. Some congruence results involving area and perimeter. 68, (1975) 282 - 283. A New Look At The "Center" Of A Triangle James E. Lightner An approach to problems associated with medians, altitudes, and angle bisectors of a triangle. 68, (1975) 612 - 615. The Converses Of A Familiar Isosceles Triangle Theorem, F. Nicholson Moore and Donald R. Byrkit Converses, the difference between necessary and sufficient conditions, use of counterexamples. 67, (1974) 167 - 170. Are Triangles That Have The Same Area and The Same Perimeter Congruent? Robert W. Prielipp Some theorems about right triangles and a counterexample. 67, (1974) 157 - 159. More About Triangles With The Same Area and The Same Perimeter Donavan R. Lichtenberg Device for decomposing a triangle with a given area and perimeter into another having the same area and the same perimeter. 67, (1974) 659 - 660. In Search Of The Perfect Scalene Triangle Bro. L. Raphael, F.S.C. Drawing a triangle which is noticeably not isosceles nor right. 66, (1973) 57 - 60. A Generalization Of Vux Triangles Charles Brumfiel A vux triangle has one angle double another. Here we consider one angle k times another. 65, (1972) 171 - 174. Using The Laboratory Approach To Relate Physical and Abstract Geometry Nancy C. Whitman Activities involving the segment determined by the midpoints of two sides of a triangle. 65, (1972) 187 - 189. The Three-Point Problem A. Day Bradley Given a triangle ABC and the angles subtended by the sides of the triangle at a point D in the same plane, find the distances DA, DB, and DC. (The problem of Pothenot.) 65, (1972) 703 - 705. An Old Stumbling Stone Revisited Robert R. Poole Trigonometry applied to the angle bisectors of an isosceles triangle. 63, (1970) 259. Vux Triangles Fitch Cheney Triangles in which one angle is double another. 63, (1970) 407 - 410. A Project In Mathematics J. Garfunkel Ten problems involving products of lengths of Cevians. 61, (1968) 243 - 248. 5 - Con Triangles Richard G. Pawley Triangles having five parts of one congruent to five (non-corresponding) parts of the other. 60, (1967) 438 - 443. Congruency Of Triangles By AAS Don Ryoti Two proofs of the theorem. 59, (1966) 246 - 247. A Scale For "Scaleneness" Evelyn B. Rosenthal Uses (perimeter - squared)/area to establish a ranking. 58, (1965) 318 - 320. A Dialogue On Two Triangles Leander W. Smith When will the perimeter and the area of a triangle be numerically equal? 57, (1964) 233 - 234. Dr. Hopkins' Proof Of The Angle Bisector Problems Sister Mary Constantia, S.C.L. A direct proof of the result that if two angle bisectors are congruent the triangle is isosceles. 57, (1964) 539 - 541. Altitudes, Medians, Angle Bisectors and Perpendicular Bisectors Of The Sides Of A Triangle Harry Schor Paper folding. 56, (1963) 105 - 106. The Problem Of The Angle Bisector Joseph Holzinger If two angle bisectors of a triangle are congruent the triangle is isosceles. The result is trigonometrically based. 56, (1963) 321 - 322. Notes On The Centroid Nathan Altshiller Court Primarily historic. 53, (1960) 33 - 35. The Use Of Congruence In Geometric Proofs Carl Bergman Using directed triangles in congruence proofs. 51, (1958) 23 - 26. Micky's Proof Of The Medians Theorem M. L. Keedy Using informal deduction to prove that the medians of an isosceles triangle are congruent. 51, (1958) 453 - 455. Congruent Triangles (Fifth Case) and The Theorem Of Lehmus Victor Thebault A proof of a congruence theorem concerning angle bisectors. 48, (1955) 97 - 98. On Certain Cases Of Congruence Of Triangles Victor Thebault Congruence theorems related to division ratios. 48, (1955) 341 - 343. Quasi-Right Triangles Adrian Struyk Triangles such that the difference of two angles is 90 degrees. 47, (1954) 116 - 118. More Than Similar Triangles Charles Salkind Triangles such that five parts of one are congruent to five (non-corresponding) parts of the other. 47, (1954) 561 - 562. A New Proof Of An Old Theorem Francis A. C. Sevier Deals with the angle bisectors of an isosceles triangle. 45, (1952) 121 - 122. Relations For Radii Of Circles Associated With The Triangle T. Freitag Circumradius, inradius, etc. 45, (1952) 357 - 360. Signed Areas Applied To "Recreations Of Geometry" H.C. Trimble Analytic approach to some triangle geometry. Danger in arguing from a figure. 40, (1947) 3 - 7. Escribed Circles Joseph A. Nyberg Geometry of the circles. 40, (1947) 68 - 70. Dynamic Geometry John F. Schact and John J. Kinsella The use of triangle and quadrilateral linkages as teaching devices. Some circle geometry. 40, (1947) 151 - 157. A New Technique In Handling The Congruence Theorems In Plane Geometry Ralph C. Miller Using constructions. 36, (1943) 237 - 239. The Congruence Theorems By A New Proof H.C. Christofferson The use of an assumption which enables one to avoid the use of superposition. 28, (1935) 223 - 227. Assuming The Congruence Theorems Joseph A. Nyberg What should be assumed and what should be proved? 24, (1931) 395 - 399. Proving The Equality Of The Base Angles Of An Isosceles Triangle Joseph A. Nyberg Need we assume the existence of an angle bisector? 22, (1929) 318 - 319. Isotomic Points Of The Triangle Richard Morris Isotomic points, Gergonne points, Nagel points. 21, (1928) 163 - 170. Circles Through Notable Points Of The Triangle Richard Morris Circles through three, four, five, and six points. 21, (1928) 63 - 71. Proof Of An Original Exercise Walter Beyer On the sides of a triangle ABC construct equilateral triangles AEC, CDB, and BFA. Show that AD, BE, and CF are concurrent. 20, (1927) 91 - 92. Interesting Work Of Young Geometers J. T. Rorer Three triangle theorems and an approximate trisection. 1, (1908-1909) 147 - 149. H.J.L. 07/15/97 email: 00hjludwig@bsu.edu home page: http://www.cs.bsu.edu/~hjludwig/ ```