CONVERGENCE OF A SPECIAL SET OF TRIANGLES

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1 Journal of Mathematical Sciences: Advances and Applications Volume Pages Available at DOI: COVERGECE OF A SPECIAL SET OF TRIAGLES Dedicated to Sherman Kopald Stein ( August 96- ) for his pleasant sense of humor and mathematical wisdom that the third author enjoyed as a colleague during the late 980s at the University of California Davis MOSHE STUPEL AVI SIGLER and JAY JAHAGIRI 3 Shannan Academic College of Education Gordon Academic College of Education Haifa Israel stupel@bezeqint.net Shannan Academic College of Education Haifa Israel avibsigler@gmail.com 3 Mathematical Sciences Kent State University Kent Ohio USA jjahangi@kent.edu Abstract An explicit expression for the convergence point of a given set of triangles is introduced leading to an open interval locus of convergence that contains the centroid of the triangles. 00 Mathematics Subject Classification: Primary: 97G40; Secondary: 97G50. Keywords and phrases: geometric convergence barycentric coordinates Ceva s theorem Pappus configuration. Received March 8 08; Revised April Scientific Advances Publishers

2 60 MOSHE STUPEL et al.. Introduction Using coordinate systems such as rectangular polar spherical and cylindrical require an origin but at times we need to locate points relative to existing points or local coordinates rather than the origin [-4]. Mobius assigned coordinates (barycentric coordinates) to the point at which the center of gravity or barycentre was located reflecting the ratio of weights attached to the ends of a given rod. He then extended this idea to a system of three weighted points forming a triangle and pointing out that the center of gravity will remain the same if the weights are increased or decreased by a common factor [5 6]. Central to barycentric coordinates are the concept of concurrency of lines in a triangle and Ceva s theorem that Given any triangle the segments from each vertex to the opposite sides of the triangle are concurrent precisely when the product of the ratios of the pairs of segments formed on each side of the triangle is equal to [5 7]. Ceva s theorem is used in the proofs of many well-known theorems including the theorem of apolean s point the theorem of Fermat s point and the theorem of the agel point [8]. Barycentric coordinates are a fundamental tool for a wide variety of applications employing triangular meshes underlining the definition of higher-order basis functions the Bézier triangle in computer aided-design and many interpolation and shading techniques in computer graphics [9 0]. In the present paper we use the convergence of a special set of triangles leading to the discovery of an explicit expression for the convergence point and the understanding that this point can be constructed using a compass and a straightedge ruler even if the convergence process is infinite. These results are then generalized to Pappus triangles and generalized Pappus triangles [-3]. Finally we obtain an interesting open interval locus of convergence that contains the centroid of the triangle.

3 COVERGECE OF A SPECIAL SET OF TRIAGLES 6. Terminology and otation The interior point D of the triangle ABC (Figure ) Figure. Representation of the point D. shall be represented by two methods. (a) Using the barycentric coordinates ( 3 ) we obtain xd yd x A + xb + 3xC ya + yb + 3yC where and (b) Using the Ceva coordinates ( α β γ) we obtain AA BB and CC which are three cevians that pass through D so BA that CB AC α β and γ where A C B A C B αβγ α 0 β 0 γ 0.

4 6 MOSHE STUPEL et al. Known are the relations between the barycentric coordinates and the Ceva coordinates 3 αβ αβ + α + αβ + α + α αβ + α + () α β γ 3 3. () 3. The Convergence Process The convergence process used here are presented in two phases. Case I: The Pappus configuration Select the points C B respectively on the sides CA BC and A AA AB so that BB CC α. A B B C C A Similarly select the points C B A respectively on the sides C A BC and A B so that A A B B CC α. AB BC CA Repeating this process ( + ) -times ( is a whole number) we arrive at the points C B A which are respectively on the sides C A B C and A so that they divide the sides of the triangle BC B A at a ratio that equals α. ote that the triangle A 0B0C0 is the original triangle ABC and the sides A 0B0 B0C0 and C 0 A0 are the sides of the original triangle ABC.

5 COVERGECE OF A SPECIAL SET OF TRIAGLES 63 The famous Pappus theorem claims that the triangles BC ABC and A have a common centroid [4]. Therefore it is clear that the triangles A BC and ABC also have a common centroid and so on and so forth up to the triangles A B C and A B C. A continuous repetition of this process leads to the convergence of the triangles A + B + C + to a single centroid point as approaches infinity and therefore we have the following: Claim. A + B + C + is the centroid point of ABC as tends to infinity. Case II: Generalization The points C B A lie respectively on the sides CA BC and AB of the triangle ABC so that BB B C α CC C A β AA A B γ (Figure ). Figure. Convergence of Pappus triangles.

6 64 MOSHE STUPEL et al. The points C B A lie respectively on the sides C A BC and A B of the triangle A BC so that B B C C A A α β γ. BC CA AB Repeating this process ( + ) -times we conclude that the points C B + A + respectively lie on the sides C A BC A B so that + B B B C + + α C C C A + + β A A A B + + γ. ext we state and prove. Claim. A B C ; a point of ABC with the S barycentric coordinates S S S ( ) ( ) αβ γ + 3 ( ) ( ) ( ) αβ γ + + βγ α + + αγ β + βγ( α + ) αβ( γ + ) + βγ( α + ) + αγ( β + ) αγ( β + ) αβ( γ + ) + βγ( α + ) + αγ( β + ) (3) or with the Ceva coordinates S S S ( ) ( ) ( ) ( ) α β + β γ + γ α + 3 β( α + ) γ( β + ) α( γ + ). (4) Proof. We transform the triangle ABC by an affine transformation to a triangle in which A B and C have the coordinates A ( 0 ) B( 0 0) and C ( 0). ext we transform the points A C B by an affine transformation to BB respective points C B A so that CC AA α β and γ. B C C A A B Repeating this process ( + ) -times we note that these affine transformations preserve the simple ratio where the triangle A BC

7 COVERGECE OF A SPECIAL SET OF TRIAGLES 65 shall be transformed into the triangle A + B + C + so that the division ratios of the corresponding sides are α β and γ as required. The affine transformation f that transforms A C B to C B is A denoted by f ( ABC) A BC (Figure 3). This is a continuous function from the convex domain ABC to the convex domain A BC that is contained in ABC. Therefore f has a fixed point S. This point S shall be the only common point of the triangles A B C Figure 3. Affine transformation f ( ABC) A B. C To derive the coordinates of the point S observe that the coordinates of A B and C are 0 α 0 β A B. γ + α + C β + β + The fixed point for f has the same barycentric coordinates in ABC and A BC. Therefore it is a solution of the following system of equation: x S ABC α α + β + A B C

8 66 MOSHE STUPEL et al. In the other words α (5) α + β + 3β +. (6) γ + β + From (5) it follows that 3 α( β + ) α S β( α + ). (5*) From (6) it follows that β( γ + ) β S γ( β + ). (6*) 3 And clearly we obtain γ S γ( α + ) α( γ + ). A simple calculation using Equation () gives αβ( γ + ) S αβ( γ + ) + βγ( α + ) + αγ( β + ). The process for S and S 3 are similar and will be omitted. ote. For given α β and γ the point S can be constructed using a compass and a straightedge ruler (though the convergence process is infinite). An accurate description of segment construction by using compass and straightedge based on expressions whose representation is by segments can be found in references [ ].

9 COVERGECE OF A SPECIAL SET OF TRIAGLES 67 ext we deal with three particular cases (Figure 4). Figure 4. The case of the middle of the segments. First particular case If α β γ then the convergence is at the centroid of ABC. Second particular case (Pappus) of If α β γ k where k( 0 < k < ) is fixed then S is the centroid ABC because αs βs γs. Third particular case (Generalized Pappus) then If BB CC kα AA kβ and kγ where k( 0 < k < ) is fixed B C C A A B αs βs γs α( kβ + ) β( kα + ) β( kγ + ). γ( kβ + ) γ( kα + ) α( kγ + ) (7) In the other words aside from the Pappus case the point S depends on k and is therefore denoted by S ( k).

10 68 MOSHE STUPEL et al. Finally we state and prove our Claim 3. The locus of ( k) S where k ( 0 < k < ) is fixed is the open interval straight-line segment bounded by two points M and M where M is the centroid of the triangle and M has the barycentric coordinates M αβ M βγ M 3 αβ + βγ + γα αβ + βγ + γα αγ. αβ + βγ + γα Proof. When k tends to zero in (7) we obtain the Ceva coordinates α β γ M which means the barycentric coordinates β γ α M ( M M ) αβ βγ αγ. 3 αβ + βγ + γα αβ + βγ + γα αβ + βγ + γα When k tends to infinity we obtain ( ) ( α β γ ) or barycentric coordinates ( ). M M M M M M 3 We now prove M and M. It is enough that S ( k) is located on the segment connecting to prove that M M M 3 M M M 3 0 S ( k) S( ) ( ) k S k 3 or αβ βγ αγ 0 αβ( kγ + ) βγ( kα + ) αγ( kβ + )

11 COVERGECE OF A SPECIAL SET OF TRIAGLES 69 or αβ βγ αγ 0. kαβγ kαβγ kαβγ We will show that S ( k) is located between M and M. We connect A to M and M and obtain the respective points of intersection D and D (Figure 5). Figure 5. The location of the point S ( k). Without loss of generality we assume that M and D are more to α the left of M and D in the other words <. β ( ) α α βk + In this case < αs ( ) < and therefore S β k β( αk + ) ( k) is located between M and M.

12 70 MOSHE STUPEL et al. References [] G. E. Martin Geometric Constructions Springer ew York (998) 3-4. [] V. Oxman M. Stupel and A. Sigler Geometric constructions for geometric optics using a straightedge only Journal for Geometry and Graphics 8() (04) [3] M. Stupel V. Oxman and A. Sigler More on geometrical constructions of a tangent to a circle with a straightedge only The Electronic Journal of Mathematics and Technology 8() (04) [4] A. Sutton Ruler and Compass: Practical Geometric Constructions Walker & Company 009. [5] P. Lidberg Barycentric and Wachspress Coordinates in two Dimensions: Theory and Implementation for Shape Transformations Uppsala Universitet U.U.D.M. Project Report February 0. [6] S. K. Stein Archimedes: What did he do Besides Cry Eureka The Mathematical Association of America Page 5 Washington 999. [7] D. A. Brannan M. F. Esplen and J. J. Gray Geometry Cambridge University Press page 75 Cambridge 004. [8] D.. V. Krishna The fundamental property of agel point: A new proof Journal of Mathematical Sciences & Mathematics Education () (07) [9] R. M. Rustamov Y. Lipman and T. Funkhouser Interior distance using barycentric coordinates Eurographics Symposium on Geometry Processing 8(5) (009) DOI: [0] J. Vince Mathematics for Computer Graphics (4th Edition) Springer London Heidelberg ew York Dordrecht 03. ISB: [] H. Eves An Introduction to the History of Mathematics Philadelphia: Saunders College Publishing 99. [] A. Kryftis A Constructive Approach to Affine and Projective Planes Ph.D. Thesis University of Cambridge 05. [3] B. D. S. McConnell A Six-Point Ceva-Menelaus Theorem 3 March [4] A. W. Goodman and G. Goodman Generalizations of the theorems of Pappus The American Mathematical Monthly 76(4) (969) g

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