The centroid of some generalized pedal configurations

Transcription

1 The centroid of some generalized pedal configurations Szilárd András Babeş-Bolyai University, Cluj Napoca, Romania... all meaning comes from analogies. Douglas Hofstadter A mathematician is a person who can find analogies between theorems, a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories, the ultimate mathematician is one who can see analogies between analogies. Stefan Banach Abstract. The main goal of this paper is to give possible generalizations, analogues of the following property: If, and are the orthogonal projections of a point to the sides A A, A A and A A of an equilateral triangle A A A, then the centroid of the triangle is the midpoint of the segment, where is the center of the triangle A A A. In the first step we extend this property to regular n-gons, regular tetrahedrons and regular n simplices. In the second part we give a general affine version for triangles and simplices. It is also our objective to analyze the possibility of using such properties in teaching problem solving strategies for students and mathematics teachers. Theorem,, 4, 6 and Conjecture and were discovered/rediscovered during a training course for mathematics teachers. Keywords: pedal polygons, pedal simplices, centroids athematical Subject Classification: 5N0, 5A0, 9740 Introduction In this note we analyze some possible generalizations of the following property: Theorem. If we denote by, and the orthogonal projections of a point to the sides A A, A A and A A of an equilateral triangle A A A, then the centroid of the triangle is the midpoint of the line segment, where is the center of triangle A A A. A A A Figure : The centroid of a podar triangle relative to an equilateral triangle address: andraszk@yahoo.com

2 This problem was studied on a teacher training course at the Babeş-Bolyai University from the perspective of the inquiry based learning. The starting point was not only the problem, but also a solution of this problem using complex numbers. The aim of the instructors was to support the participants secondary school mathematics teachers in experimenting, developing and proving generalizations of this property. This approach was used in order to prove that inquiry based learning can also be used in problem solving activities and in the framework of the existing curricula. The main tools for experimenting were dynamic geometry softwares eogebra, eonext and Cabri and the teachers were working in groups. In the first round each group had to present some ideas about the possible generalizations, in the second round each group had to formulate conjectures and had to experiment his own conjectures and in the final round they had to prove the conjectures which seemed to be valid. Conjectures and proofs As a first step we can replace the equilateral triangle with something more general: an arbitrary triangle, a regular polygon, a regular tetrahedron or a regular simplex. r we can use some more general projections in constructing the pedal triangle. In order to formulate some more general properties we recall a proof for theorem. Proof. Without loosing generality we can assume that is in the plane of the triangle A A A, and the circumscribed circle is of radius. Let s consider as the origin of the coordinate system and A as the X axis. The complex numbers corresponding to the vertices A, A and A are a ε, a ε and a ε, where ε and ε. Due to our assumptions, m is easy to find, because it s real part is and it s imaginary part is the same as m s. So m + m m. To calculate m we ll use a counterclockwise rotation of angle α 4π. This rotation transforms the point to the orthogonal projection of the point Qε m to the side A A. Hence we have m ε ε m ε m +, and so By the same argument we deduce From these relations we get which expresses the desired property. m ε + m ε m. m ε + m ε m. m + m + m m, The above presented proof suggests that something similar holds also for a regular n-gon. A little experience with a dynamic geometry software helps us to formulate the following property: Theorem. If we denote by i i n the orthogonal projections of a point to the sides A i A i+ i n and A n+ A of a regular n-gon A A A... A n and by the center of the polygon, then the centroid of the n-gon... n is the midpoint of the line segment.

3 A A A 7 A A 6 A 4 A5 Figure : The centroid of a podar polygon relative to a regular n-gon Proof. First we consider the case when n is odd. The vertices of the regular n-gon are represented by a j ε j cos jπ n If n k +, we have A k A k+ Y and hence Using rotations again we can deduce m j ε k j cos + sin jπ n, j n. kπ m k cos k + + m m. kπ k + + m εk j m ε k j for j k +. But ε ε k ε and hence we have the following relations: These relations and the n where m j ε k+j+ cos v0 kπ k + + m εj+ m ε v 0 equality imply n m j m. j for j k +. If n is even n k we take a j z 0 ε j z 0 cos jπ jπ + i sin, j n, n n z 0 cos π k + i sin π k

4 and we obtain m k cos k π k + m m, m j ε k j k π cos + m εk j m ε k j k for j k. These relations imply n m j m and hence the proof is completed. j Remark. If n k, the points corresponding to the complex numbers mj+m j+k are on the axis of symmetry parallel to A j A j+. So the relation n m j k j j m j + m j+k m means that the centroid of the k-gon determined by the projections of to the these axis of symmetry is the midpoint of. In this case the property can be viewed as theorem with a degenerated regular k-gon which degenerates into the center of symmetry. The previous remark suggests that a similar property holds if the points,,..., n are the projections of to the axis of symmetry. To explore this possibility and to formulate a proper conjecture we can construct figures using a geometry software. Finally we obtain the following property: Theorem. The centroid of the n-gon determined by the orthogonal projections of a point to the axis of symmetry of a regular n-gon is the midpoint of is the center of symmetry. A A A 5 A A 4 Figure : The centroid of a podar polygon relative to the symmetry axes of a regular n-gon 4

5 Proof. The method and the calculations are almost the same. We can assume that one of the symmetry axis is the Y axes, so the real part of the projections affix is 0 while the imaginary part is the same as the imaginary part of the projected point s affix. This implies that if we repeat all the calculations instead of cos k π k or cos kπ k+ we have 0. Remark. All groups discovered/rediscovered independently theorem while theorem was observed only by one group when they analyzed their proof. In order to generalize the previous results to higher dimensions as a first step we analyze the case of a regular tetrahedron. Theorem 4. Let A A A A 4 be a regular tetrahedron with center, i i 4the orthogonal projections of a point to it s faces and Q i i 6 the projections to it s sides. The following two statements are true: a The centroid of the tetrahedron 4 is on and satisfies. b The centroid of the system Q Q Q Q 4 Q 5 Q 6 is on and satisfies. A A Q Q A 4 A 4 A Q 5 Q A 4 Q 6 Q 4 A A Figure 4: The centroid of a podar tetrahedron relative to the faces and the sides of a regular tetrahedron Proof. First we prove that the two statements are equivalent. Due to theorem. we have see figure 4: + Q 4 + Q 5 + Q 6, + Q 4 + Q + Q, + Q + Q 5 + Q and Q + Q + Q 6, where Q A A, Q A A 4, Q A A, Q 4 A 4 A, Q 5 A A 4, Q 6 A A, AB means the vector from A to B, and i are the centroids of the faces. From these equalities we deduce 4 i i 5 6 Q i i

6 because 4 i 0 and so i. This relation implies the equivalence of the two statements. In order to prove the first statement we consider a regular tetrahedron with vertices A 0, 0, 6 6, 0, A, 6, 0 A, equations of the faces are: Using the formula, A 4 0,, 0 and we calculate the coordinates of the points i. The A A A : y + z 0, A A A 4 : x 6y z + 0, A A A 4 : x + 6y + z 0 and A A A 4 : z 0 x x 0 A A x 0 + B y 0 + C z 0 + D A + B + C, which gives the x coordinate of the orthogonal projection of the point x 0, y 0, z 0 to the plane with equation A x + B y + C z + D 0 we obtain: x x 4 x 0, x x 0 x y 0 + z 0 7 x x 0 x 0 6 y 0 z From these equalities we can easily deduce the relation x + x + x + x 4 4 x 0. A similar calculation shows the same relation for the y and the z coordinates, hence lies on P and satisfies the desired equality. This completes the proof. Based on this property we can formulate the following conjecture for the n-dimensional euclidian space: Conjecture. If we denote by k the centroid of the system determined by the orthogonal projections of a point to the k-dimensional faces of a regular n-simplex A A A...A n A n+, than k and satisfies k k n, k n, where is the center of the simplex. Remark. Theorem 4 was discovered/rediscovered by all the groups while conjecture was formulated without proof only by one group. After an analysis of the proof of theorem 4 all groups formulated the necessity of a different approach in attacking the higher dimensional problem. This conjecture can be proved using the same ideas as in the proof of theorem 4, but the calculations are more complicated. In order to simplify the proof of this conjecture first we try to give an affine version of the dimensional property and then by extending this affine version to higher dimensions we find a property which is more general than conjecture and admits a simpler proof. For this we need to observe that in a regular polygon polyhedra or simplex the segment joining the centroid of the polygon with the midpoint i of a side the centroid of a face is perpendicular to this side face. Hence the construction of the orthogonal projection of to a side d can be viewed as the construction of a projection in the direction i. This can be done for an arbitrary triangle or even for an arbitrary polygon, so we can formulate the following conjectures: and, 6

7 A A A Figure 5: The centroid of a generalized podar triangle relative to an arbitrary triangle Conjecture. In the triangle A A A A A, A A and A A are the midpoints of the sides and A A A {}. If for an arbitrary point we consider the points A A, A A and A A such that, and, then the centroid of the triangle is the midpoint of the segment. Conjecture. Denote by,,..., n the midpoints of the sides A A, A A,..., A n A in the polygon A A...A n and by the centroid of the polygon. If for an arbitrary point we consider the points i A i A i+, i n such that i i, i n, then the centroid of the polygon... n is the midpoint of the segment. Using a dynamic geometry software we can explore the validity of these conjectures. Conjecture seems to be true but unfortunately Conjecture is not true for all polygons. Remark 4. It is an open question to assure conditions for which conjecture becomes a theorem. ne of the groups observed that conjecture becomes a theorem for n 4 if A A A A 4 is a trapezoid. Proof of conjecture. If we denote by the corresponding small letter the position vector of each point, we have the following relations: o a +a +a, o a +a, o a +a and o a +a. is an arbitrary point in the plane so there exist α, α, α R such that m α a + α a + α a and α + α + α. Since A A there exists λ R with the property m λ a + λ a, hence the condition can be expressed as with c R. This implies m m co o α a + α λ a + α + λ a c a 6 a 6 a. If the origin of the position vectors is not in the same plane as the vertices of the initial triangle, than a, a and a are linearly independent vectors, so we obtain the following system α c α λ c 6 α + λ c 6 and so m m α o o α + α a + α + α a. 7

8 Using a similar argument we obtain m α + α a + α + α a and m α + α a + α + α a so g m + m + m m + o. Analyzing theorem 4 and conjecture we can formulate the following property: Theorem 5. Zsolt Szilágyi, Szilárd András Consider the n-simplex A 0... A n, an arbitrary point and the centroid of the simplex. Denote by i0...i k the centroid of the face A i0... A ik and by i0...i k the intersection of the face A i0... A ik with the line drawn through and parallel to i0...i k A ik+...a in. If k denotes the centroid of the system i 0...i k, where k is fixed and i 0... i k takes all possible values, then for each k {,,..., n } the points,, k are on the same line and k k n. A X X 4 4 A A 4 X A Figure 6: The centroid of a podar tetrahedron relative to an arbitrary tetrahedron X Proof. Denote by the corresponding small letters the position vectors in R n of the points. is an arbitrary point in R n, so there exist α 0,...,α n R such that n α i and m α i a i. i0 If m : m i0...i k k c j a ij with k c j is the intersection of the face A i0... A ik with the parallel line j0 j0 to i0...i k A ik+... A in, and o i0...i k k+ i0 a ij is the centroid of the face A i0... A ik, then the condition j0 8

9 of parallelism is: m m c where λ j R, for all j {k +,...,n} and α ij c j a ij + j0 jk+ jk+ jk+ α ij a ij c λ j a ij o i0...i k, λ j. This can be written as jk+ so we deduce λ j αi j c, for k + j n, and c j α ij + c obtain c n jk+ α ij, and so m m i0...i k j0 k+ λ j a ij k + j0 [ α ij + ] k + α i k α in a ij. The number of the k dimensional faces in an n-simplex is C k+ n+, hence g k C k+ n+ C k+ n+ i C n+,k+ m i0...i k i C n+,k+ j0 a ij,, for 0 j k. From these relations we [ α ij + ] k + α i k α in a ij, where C n+,k+ denotes the set of all combinations of order k + formed from the set {0,,..., n+ }. The number of combinations i for which l {i 0,...,i k }, where l is a fixed index, is Cn k while the number of combinations i for which l {i 0,...,i k } and j {i k+,...,i n } is Cn k. This implies g k C k Cn+ k+ n α l + Cn k k + α j a l l0 j l [ ] Cn k α l + Ck n k + α l a l The centroid is o n+ But n l0 a i, so i0 k Cn+ k+ l0 k + n + α l + l0 k n α l + l0 l0 l0 α l n+ a l, hence n k nn + α l n k nn + a l. n + + k n α l + k nn + + k n α l a l a l k k n, which completes the proof. n k nn + a l. 9

10 Remark 5. Theorem 5 proves Conjecture, because in a regular simplex i0...i k A ik+... A in is orthogonal to the face A i0... A ik. At the training course this theorem and it s proof was presented by the instructors. Examining theorem the following natural question arise: can we replace the midpoints of the faces by other points? The following theorem answers this question. Theorem 6. Consider a triangle A A A and the real numbers w, w, w with w + w + w. Let be the point with barycentric coordinates w, w, w and an arbitrary point in the plane of the triangle. If we construct the points A A, A A and A A such that A, A and A, then the centroid of the triangle coincides with the centroid of the triangle P, where the barycentric coordinates of P relative to are w, w, w. Remark 6. If w w w, then P is the centroid of, so the above theorem reduces to conjecture. A B P B A A Figure 7: The centroid of a generalized podar triangle relative to an arbitrary triangle B Proof. If γ, γ, γ denote the barycentric coordinates of, then and m γ a + γ a + γ a o w a + w a + w a, where a, a, a are the position vectors of the vertices and m, o the corresponding position vectors of the points and. A A, so there exists λ R such that m λ a + λ a. The condition A can be expressed as m m co a, with some c R. This implies λ a + λ a γ a γ a γ a cw a + w a + w a a If we suppose that the starting point of the position vectors is outside the plane A A A, then a, a, a are linearly independent, hence implies γ cw, λ γ cw and λ γ cw. From these relations we obtain w w m γ + γ a + γ + γ a. w + w w + w By a similar reasoning we deduce w m γ + γ m w + w w γ + γ w + w w a + γ + γ a + a, 4 w + w a. 5 γ + γ w w + w 0

11 From, 4 and 5 we deduce m w + m w + m w m + o, thus which is the desired property. m + m + m m + o + p, Remark 7. Theorem 6 can be extended also to higher dimensional simpleces. Remark 8. It would be interesting to find an affine version which generalizes all the previous properties including theorem too. Concluding remarks From scientific point of view the training course was fruitful because the participants discovered new properties of the studied configurations and they developed new generalizations, which clarify some aspects of the problem. Remark 4 and remark 8 shows t hat interesting questions arose, they can constitute a starting point for further study. From a teaching point of view the main aim of the training course was achieved, the participants got familiar with the inquiry based methods, they realized that a deeper inquiry of the problems can constitute a solid motivational base for further individual study. They also perceived that in many cases the understanding of the background implicitly requires a deeper inquiry and this can be done even in the traditional framework and context. Some participants pointed out that the major inference of this training course was that they realized the importance of simultaneous alternatives in solving a problem and the importance of selecting the tools they use in designing the proofs of well known properties. The participants emphasized that a major task in preparing and designing an inquiry based lesson is the selection of a sufficiently rich problem. Acknowledgements This paper is based on the work within the project Primas. Coordination: University of Education, Freiburg. Partners: University of enève, Freudenthal Institute, University of Nottingham, University of Jaen, Konstantin the Philosopher University in Nitra, University of Szedged, Cyprus University of Technology, University of alta, Roskilde University, University of anchester, Babeş-Bolyai University, Sør-Trøndelag University College. The author wishes to thank his students and colleagues attending the training course organized by the Babeş-Bolyai University in the framework of the FP7 project PRIAS. The author was partially supported by the Hungarian University Federation of Cluj Napoca. References [] Szilárd András: About the centroid of podar polygons, ctogon athematical agazine, 900:, [] Szilárd András, Zsolt Szilágyi: eometria II in hungarian, Státus, Csíkszereda, 006 [] Titu Andreescu, Dorin Andrica: Complex Numbers from A to...z, Birkhäuser Boston, 005 [4] arian Dincă, arcel Chiriţă : Aplicaţii ale numerelor complexe în geometrie, Editura All Educaţional, 996 Promoting inquiry in athematics and science education across Europe, rant Agreement No. 4480