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Forum Geometricorum Volume 6 (2006) 229–234.

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FORUM GEOM ISSN 1534-1178

On Some Theorems of Poncelet and Carnot Huub P.M. van Kempen

Abstract. Some relations in a complete quadrilateral are derived. In connection with these relations some special conics related to the angular points and sides of the quadrilateral are discussed. A theorem of Carnot valid for a triangle is extended to a quadrilateral.

1. Introduction The scope of Euclidean Geometry was substantially extended during the seventeenth century by the introduction of the discipline of Projective Geometry. Until then geometers were mainly concentrating on the metric (or Euclidean) properties in which the measure of distances and angles is emphasized. Projective Geometry has no distances, no angles, no circles and no parallelism but concentrates on the descriptive (or projective) properties. These properties have to do with the relative positional connection of the geometric elements in relation to each other; the properties are unaltered when the geometric figure is subjected to a projection. Projective Geometry was started by the Grecian mathematician Pappus of Alexandria . After more than thirteen centuries it was continued by two Frenchmen, Desargues and his famous pupil Pascal. The latter one published in 1640 his well-known Essay pour les coniques. This short study contains the well-known hexagrammum mysticum, nowadays known as Pascal’s Theorem. Meanwhile, the related subject of perspective had been studied by architects and artists (Leonardo da Vinci). The further development of Projective Geometry was about two hundred years later, mainly by a French group of mathematicians (Poncelet, Chasles, Carnot, Brianchon and others). An important tool in Projective Geometry is a semi-algebraic instrument, called the cross ratio. This topic was introduced, independently of each other, by M¨obius (1827) and Chasles (1829). In this article we present an (almost forgotten) result of Poncelet [4], obtained in an alternative way and we derive some associated relations (Theorem 1). Furthermore, we extend a theorem by Carnot [1] from a triangle to a complete quadrilateral (Theorem 3). Publication Date: September 25, 2006. Communicating Editor: Floor van Lamoen. The author thanks Floor van Lamoen and the referee for their useful suggestions during the preparation of this paper.

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We take as starting-point the theorems of Ceva and Pappus-Pascal. The first one is a close companion of the theorem of the Grecian mathematician Menelaus. In the analysis we will follow as much as possible the purist/synthetic approach. It will be shown that this approach leads to surprising results derived along unexpected lines. 2. Proof and extension of a Theorem by Poncelet Theorem 1. Let the diagonal points of a complete quadrilateral ABCD be P, Q and R. Let the intersections of PQ with AD and BC be H and F respectively and those of PR with CD and AB be G and E respectively (Figure 1). Then AE BF CG DH · · · =1, (1) EB F C GD HA AP CG DP BE · · · =1, (2) P C GD P B EA BP DH AP CF · · · =1. (3) P D HA P C F B R

C

G

F

D P H

Q

A

E

B

Figure 1

Proof. We apply the Pappus-Pascal theorem to the triples (Q, A, E) and (R, C, F ) and find that in triangle ABC the lines AF , BP and CE are concurrent so that by Ceva’s theorem AE BF CP · · = 1. (4) EB F C P A Similarly with the triples (Q, G, C) and (R, H, A) we find CG DH AP · · = 1. (5) GD HA P C

On some theorems of Poncelet and Carnot

231

Relation (1) immediately follows from (4) and (5). Again in the same way with triples (E, B, Q) and (H, D, R) we find that AE BP DH · · = 1. (6) EB P D HA (2) follows from (5) and (6), and (3) follows from (4) and (6).  Poncelet [4] has derived relation (1) by using cross ratios. We now consider a special case of Theorem 1, taking a convex quadrilateral ABCD in which AB + CD = BC + DA, so that it is circumscriptable (Figure 2). Let E  , F  , G and H  be the points of tangency of the incircle with AB, BC, CD and DA respectively. Clearly a relation similar to (1) holds: AE  BF  CG DH  · · · = 1. (7) E  B F  C G D H  A

R

C

G

G F

D F

H

P

M

H

Q

A

E

E

B

Figure 2

It is well known [5] that the point of intersection of E G and F  H  is P . This can be seen for a general quadrilateral with an inscribed conic from subsequent application of Brianchon’s theorem to hexagons AE BCG D and BF  CDH  A. See for instance [2, p.49]. This raises the questions whether or not a relation similar to (1) will hold. We will examine this problem by using Ceva’s theorem.

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H. P. M. van Kempen

3. Further Analysis We start with a given quadrilateral ABCD where E and G are arbitrary points on the lines AB and CD respectively. We then construct points F1 and H1 on BC and AD respectively such that (1) holds. We can do so by the following construction (Figure 3). G

C

D

S P H1

K

F1 T

A

E

B

Figure 3

First we consider the triangles ABC and ADC. Let T = CE ∩ BD and F1 = BC ∩ AT . By Ceva’s theorem we have in triangle ABC AE BF1 CP · · = 1. (8) EB F1 C P A Now if S = AG ∩ DB and H1 = AD ∩ CS, then Ceva’s theorem applied to triangle ADC gives CG DH1 AP · · = 1. (9) GD H1 A P C By multiplication of (8) and (9) we find the desired equivalence of (1). Theorem 2. If in the quadrilateral ABCD the points E, F1 , G and H1 lie on AB, BC, CD and DA respectively such that S = AG ∩ CH1 and T = AF1 ∩ CE lie on BD, then the points A, E, F1 , C, G and H1 lie on a conic and K = EG∩F1 H1 lies on BD. Proof. Here we have to switch to the field of Projective Geometry. We will use the cross ratio of pencils in relation to the cross-ratio of ranges. These concepts are extensively described by Eves [3]. Now consider the two pencils (AH1 , AG, AF1 , AE) and (CH1 , CG, CF1 , CE) in Figure 3. We have the cross-ratio equality between ranges and pencils:

On some theorems of Poncelet and Carnot

233

A(H1 , G; F1 , E) = (D, S; T, B) = (S, D; B, T ) = C(H1 , G; F1 , E).

(10)

From this equality we see that A, E, F1 , C, G and H1 lie on a conic. Applying Pascal’s theorem to the hexagon AF1 H1 CEG we find that the diagonal BD is the Pascal line and consequently the points S, K and T are collinear.  By using triangle ABD and triangle CBD instead of triangle ABC and triangle ADC as above, we can also construct F2 and H2 such that relation (1) holds (Figure 4). Now we apply Theorem 2, finding that B, F2 , G, D, H2 and E lie on a conic. Using Pascal’s theorem for the hexagon BGEDF2 H2 we find that L = EG ∩ F2 H2 lies on AC. G

C

D

L

F2

H2 P

A

E

B

Figure 4

With the help of Theorem 2 we prove an extension of Carnot’s theorem in [1] for a triangle to a complete quadrilateral. Theorem 3. If in the quadrilateral ABCD the points E, F , G and H lie on AB, BC, CD and DA respectively and EG and F H concur in P = AC ∩ BD, then (1) is satisfied if and only if there is a conic inscribed in quadrilateral ABCD, which touches its sides in the points E, F , G and H. Proof. Assume that relation (1) holds. By Theorem 2 we know that BF GDHE and AEF CGH lie on two conics. Let V = DE ∩ BH and W = DF ∩ BG. First we apply Desargues’ theorem to triangle GF C and triangle EHV (Figure 5). The lines GE, F H and CV concur in P . This means that the intersection points of the corresponding sides are collinear. So U = GF ∩ EH, B = F C ∩ HV and D = GC ∩ EV are collinear. Next, consider the unique conic Γ through E, F , G and H which is tangent to CD at G. We examine the direction of the tangent to Γ at the point H. Therefore we consider the hexagon GGEHHF . We find that

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H. P. M. van Kempen

C

G D

W

H

P =K F

V

A

E

B

Figure 5

GE ∩ HF = P and GF ∩ EH = U . Since both P and U are collinear with B and D, the line BD is the Pascal line. This means that the tangents to Γ at G and H intersect on BD, which implies that AD is the tangent to Γ at H. In the same way we prove that the lines AB and BC are tangent to Γ at E and F respectively, which proves the sufficiency part. Now assume that a conic is tangent to the sides of quadrilateral ABCD at the points E, F , G and H. Note that of course EG and F H intersect in P , as stated earlier. With fixed E, F and G there is exactly one point H∗ on AD such that the equivalent version of relation (1) holds. By the sufficiency part this leads to a conic tangent to the sides at E, F , G, and H∗ . As these two conics have three double points in common, they must be the same conic. This leads to the conclusion that  H and H ∗ are in fact the same point. This proves the necessity part. Applying Theorem 3 to the results of Theorem 2 we find Corollary 4. If in the quadrilateral ABCD of Theorem 2 the lines EG and F1 H1 concur in P , where P = AC ∩ BD, then F1 H1 of Figure 3 and F2 H2 of Figure 4 coincide. References [1] [2] [3] [4] [5]

L. N. M. Carnot, Essai sur la th´eorie des transversals, Paris 1806. R. Deaux, Compl´ements de G´eom´etrie plane, De Boeck, Brussels 1945. H. Eves, A survey of Geometry, Allun & Bacon, Boston 1972. J. V. Poncelet, Trait´e des propri´et´es projectives des figures, Bachelier, Paris 1822. P. Yiu, Euclidean Geometry, (1998), available at http://www.math.fau.edu/yiu/Geometry.html.

Huub P.M. van Kempen: Prins Mauritsplein 17, 2582 NC Den Haag, The Netherlands E-mail address: [email protected]

On Some Theorems of Poncelet and Carnot - Forum Geometricorum

Sep 25, 2006 - analysis we will follow as much as possible the purist/synthetic ... ABCD in which AB + CD = BC + DA, so that it is circumscriptable (Figure. 2).

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