Polygons with concurrent medians

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1 Polygos with cocurret medias Joh P. Steiberger epartmet of Mathematics, U avis jpsteib@math.ucdavis.ca November 2, 2003 For Thomas achoff o the occasio of his 65th birthday. bstract The medias of a odd-sided polygo are the lies joiig vertices to the middle of their opposite sides. Triagles have cocurret medias but odd polygos with more sides are ot always so lucky. I this paper we show that if 1 medias of a odd -go are cocurret, the the remaiig media is also cocurret with the others. quivaletly, for odd ay cosecutive vertices (i a geeral positio) uiquely determie i every case a -go with cocurret medias. Thus petagos with cocurret medias, for istace, are determied by choosig four vertices at radom i the plae. short history I the fall of 1998 I had the chace of beig the teachig assistat for Thomas achoff s course Fudametal problems of Geometry. The course format relied heavily o a web-based discussio which I helped Tom moitor. e of the problems Tom put up for the studets was simply the followig: which petagos have cocurret medias? t that poit Tom himself did t eve kow the aswer (if such a classificatio questio ever has a right aswer), which was typical of his mathematical ouverture d esprit. However, Tom did tell me he supposed those petagos were exactly the liear deformatios of the regular petago. I agreed with him ad gave it o more thought; the studets would figure out the details for us. The a few days later I awoke sweaty ad clammy-haded; I d had a ightmare featurig a petago with cocurret medias ad a pair of parallel sides. This 1

2 1/4 1 x x Figure 1: petago with cocurret medias ad parallel sides. ightmarish petago (show i Fig. 1) could ot be the liear image of the regular petago sice liear trasformatios preserve parallelism. Without havig see the petago himself, Tom told me to show it i class. Whe I d fiished drawig the costructio I had to ru to meet aother appoitmet, but with the petago still o the board ad lookig somewhat dismayed at the appearace of this odd case, Tom said: You re makig a very dramatic exit, do you realize? It was udoubtedly, thaks to Tom, the high poit of my career as a teachig assistat. We thus kew the class of petagos with cocurret medias was larger tha the set of liear deformatios of the regular petago, but without kowig exactly how much larger. ur iitial observatios seemed to idicate there were seve degrees of freedom i all: six degrees of freedom oly accouted for all the traslates of liear deformatios of the regular petago (two degrees of freedom for the origi plus four degrees of freedom for the trasformatio), whereas eight degrees of freedom seemed too much. Ideed, from four vertices,, ad or a total of eight degrees of freedom oe could ifer the poit where medias crossed, ad from there a uique positio for the last vertex such that the lie through ad bisected the side (see Fig. 2), but there seemed o a priori reaso why the lie through ad would bisect the side. I fact, our less-tha-approximate had sketches strogly suggested this was t the case. Udeterred, oe of the studets, avid Ziff, decided to model the costructio o Geometer s Sketchpad. What he discovered was that the lie through ad always bisected the side. We were at oce surprised ad delighted by Ziff s discovery: it meat that we fially kew what had to be proved (aturally Geometer s Sketchpad was oly a source of empirical evidece ad it still befell to us to fid a real proof). 2

3 Figure 2: etermiig the last vertex of a petago with cocurret medias from its first four vertices. It is ot obvious whether ad are equidistat from the lie through ad. achoff foud a first proof of Ziff s observatio usig vector geometry ad I later devised a uclidea proof (oe of the studets foud a proof). Ukow to us at the time, G.. Shephard had just recetly metioed the same result as a problem for the reader i a paper of his, the referece of which I have mometarily lost. I do ot kow ay other refereces to this result. I this paper we shall give a ew simple proof of Ziff s observatio which applies ot just to petagos but to odd -gos i geeral. ur mai result has two equivalet formulatios: Theorem 1 If a odd -go has 1 cocurret medias the the last media is cocurret with the rest. Theorem 2 If is odd ad P 1,..., P /2 +1 are poits i geeral positio the there exist uique poits P /2 +2,..., P such that the polygo P 1 P 2 P has cocurret medias. ( set of poits is i geeral positio if its distributio i the plae is essetially radom.) 3

4 R L 2 Q L 1 P Figure 3 We shall first show that Theorem 2 is a cosequece of Theorem 1. We shall the give a proof of Theorem 1 for the case = 5, which the reader should have o problem geeralizig to other values of. Notatio: if ad are two poits the deotes the lie through ad, deotes the 2-by-2 determiat of the poits ad cosidered as vectors, ad deotes the segmet from to. The media through a vertex V is always deoted M V, eve if the lie has o label o the accompayig diagram (which we do to allow us to reduce clutter). If L is a lie ad S is a segmet, the L bisects S ad L is a media of S mea the same, amely that L cotais the midpoit of S. reductio I this sectio we show that Theorem 2 reduces to Theorem 1. It is sufficiet to show that every group of vertices uiquely determies a -go with 1 cocurret medias. We use the followig elemetary lemma: Lemma 3 (See Fig. 3) Let L 1 ad L 2 be two o-parallel lies itersectig at a poit ad let P be a poit ot o L 2. Let Q be the itersectio of the parallel to L 2 through P with L 1 ad let R be the itersectio of L 2 with the parallel to P through Q. The R is the uique poit o L 2 that is the same distace to L 1 as P ad which is o the opposite side of L 1 from P. Proof: bviously there exists oly oe poit o L 2 which is the same distace to L 1 as P ad which is o the other side of L 1 from P. ut R is o L 2 ad by costructio P, Q, R, ad form a parallelogram whose diagoal Q coicides with L 1, meaig 4

5 G F F Figure 4 P ad R are equidistat to L 1 ad o opposite sides. s a corollary the poit foud i the costructio of Fig. 2 is the uique poit such that that the lie through ad the midpoit of bisects the segmet. Thus whe = 5 ay 4 = vertices i geeral positio uiquely determie a petago with 4 = 1 cocurret medias (amely the medias M, M, M ad M ). So Theorem 2 reduces to Theorem 1 whe = 5. Now let = 7. Sice = 5 we wish to show that ay geeral placemet of 5 cosecutive vertices,,, ad uiquely determies a 7-go with 6 cocurret medias. The poit of itersectio of the medias is give by the itersectio of medias M ad M, which are kow from the iitial vertices. The rest of the costructio proceeds aalogously to the case = 5 (see Fig. 4) util all vertices of the heptago have bee determied, at which poit exactly 6 = 1 medias are kow to pass through directly by costructio. I geeral, if P 1 P 2... P 2 +1 are cosecutive vertices of a odd -go, the the medias through P 1 ad P 2 +1 are kow sice the opposite side of P 1 is P 2 P 2 +1 ad the opposite side of P 2 +1 is P 1 P 2. Therefore the poit of iter- 5

6 R P Q Figure 5 sectio of the medias will be kow. Now each remaiig media of the -go will either go through oe of the vertices P 2,..., P 2 or else through a midpoit of oe of the kow sides; assumig the medias are cocurret through, we thus kow all their positios. The by repeated applicatios of Lemma 3 we may successively determie uique positios for vertices P 2 +2,..., P such that side P k P k 1 is bisected by M for Pk k (the oly remaiig media which may ot be cocurret through is thus media M P ). Thus we have 1 cocurret 2 medias ad Theorem 1 ca be applied. How to prove Theorem 1 ur ew shiy proof of Theorem 1 rests o the followig simple result: Lemma 4 Let be the origi of the artesia plae, ad let P, Q ad R be three poits. The P is a media of the segmet QR if ad oly if P Q = RP. Proof: First assume P is a media of QR. Sice Q ad R are at equal distace from P, the parallelograms spaed by P ad Q ad spaed by P ad R have equal area (see Fig. 5). So the two determiats P Q ad RP are equal i absolute value, but they must also be the same sig sice Q ad R are o opposite sides of P. Therefore P Q = RP. Similarly if P Q = RP the Q ad R must be at a equal distace from P ad o opposite sides of P give that the determiats have equal sig. Therefore P is a media of QR. We shall ow just give a proof of Theorem 1 for the case = 5. Limitig ourselves to = 5 allows us to focus o the ideas rather tha o the otatio. However, the 6

7 proof s structure is so trasparet that the reader should have o problem geeralizig it to other (odd) values of. The reader may also wish to check that the same proof applies for the case = 3, i which case Theorem 1 simply states that the three medias of a triagle are cocurret. Proof of Theorem 1 for = 5: Let be a petago i which all medias except possibly for M itersect at a poit, which we assume WLG to be the origi of the artesia plae. y Lemma 4 we have that = (sice is a media) = (sice is a media) = (sice is a media) = (sice is a media) (1) d therefore = = = = which implies that is a media of by Lemma 4. 7

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