VVarignon’s and Wittenbauer’s parallelograms
Yuriy Zakharyan
Abstract.
In this paper the concept of homothetic parallelogram is in-troduced. This concept is a generalization of Varignon’s and Witten-bauer’s parallelograms of an arbitrary quadrangle, whose diagonals arenot parallel. A formula for the area and perimeter of a homothetic par-allelogram for the case when quadrangles are not crossed is obtained.The fact that homothetic parallelograms are similar to one another andare in perspective from diagonals intersection point is proved.
Mathematics Subject Classification (2010).
Keywords.
Parallelogram, homothety, perspective, quadrangle, Varignon’stheorem, Wittenbauer’s theorem.
1. Introduction
We start with an arbitrary quadrangle
ABCD , diagonals of which are notparallel. Let its diagonals intersect at point O . In this article we will focuson two related theorems. Theorem (Varignon).
Midpoints of the sides of an arbitrary quadrangle forma parallelogram. It is called Varignon’s parallelogram (Fig.1-a). [1, p.53]
Theorem (Wittenbauer).
Let the sides of an arbitrary quadrangle be dividedinto three equal parts. Lines that join dividing points near its vertices form aparallelogram. It is called Wittenbauer’s parallelogram (Fig.1-b). [2, p.216] a r X i v : . [ m a t h . HO ] M a y Yuriy ZakharyanFig.1 Varignon’s and Wittenbauer’s parallelogramsIt should be noticed that the quadrangle can be convex, re-entrant (Fig.2-a)or crossed (Fig.2-b). [1, p.52]Fig.2 Varignon’s parallelogram of different quadranglesThere are a few statements related to Varignon’s and Wittenbauer’s paral-lelograms.
Statement.
Let a quadrangle be convex or re-entrant and its area be S ABCD .The area of Varignon’s parallelogram is S ABCD . The area of Wittenbauer’sparallelogram is S ABCD . In fact, for a crossed quadrangle these areas are propotional to the differencebetween triangle areas with the same coefficients.
Statement.
Varignon’s and Wittenbauer’s parallelograms are rectangles if andonly if the quadrangle diagonals are perpendicular (Fig.3-a). Varignon’s andWittenbauer’s parallelograms are rhombuses if and only if the quadrangle di-agonals are equal (Fig.3-b). arignon’s and Wittenbauer’s parallelograms 3Fig.3 Varignon’s rectangle and rhombus
2. Generalization
Varignon’s and Wittenbauer’s theorems are obviously related. In both the-orems the quadrangle sides are divided into equal parts, dividing points arejoined by lines and these lines form parallelograms. This similarity allows usto generalize these theorems.Let us divide the quadrangle sides into n ∈ N , n ≥ A λB (or A B ( λ )) mean homothetic tranformation of the point B with cen-ter A and ration λ . Let us formulate the first theorem. Theorem 1. ∀ λ ∈ R lines A λD A λB , B λA B λC , C λB C λD , D λC D λA form a parallel-ogram. Let us call it a homothetic parallelogram of the ABCD with ratio λ . Yuriy ZakharyanFig.5 Homothetic parallelograms, λ = , Proof.
From homothety it follows: (cid:12)(cid:12) BB λA (cid:12)(cid:12) | BA | = | λ | (cid:12)(cid:12) BB λC (cid:12)(cid:12) | BC | = | λ | Moreover, ∆
ABC, ∆ B λA BB λC have a common angle. Therefore ∆ ABC ∼ ∆ B λA BB λC and AC || B λA B λC . [2, p.8]By analogy, AC || D λA D λC BD || A λB A λD BD || C λB C λD Finally, B λA B λC || D λA D λC A λB A λD || C λB C λD (cid:3) From the proof of Theorem 1 we have
Corollary 1.
The sides of a homothetic parallelogram are parallel to the quad-rangle diagonals. Also, homothetic parallelograms are similar to each other.
Corollary 2.
The homothetic parallelogram is a rectangle if and only if thequadrangle diagonals are perpendicular. The homothetic parallelogram is arhombus if and only if the quadrangle diagonals are equal.Remark . Varignon’s parallelogram is a homothetic parallelogram with ratio λ = . Wittenbauer’s parallelogram is a homothetic parallelogram with ratio λ = .The homothetic parallelogram with ratio λ = 1 is the diagonals intersectionpoint (Fig.6-a), the homothetic parallelogram with ratio λ = 0 is a limitingparallelogram (Fig.6-b).arignon’s and Wittenbauer’s parallelograms 5Fig.6 Homothetic parallelograms, λ = 1 , Remark . From this moment let a homothetic parallelogram with ratio λ be K λ L λ M λ N λ , where K λ = A λB A λD ∩ B λA B λC L λ = B λA B λC ∩ C λB C λD M λ = C λB C λD ∩ D λA D λC N λ = D λA D λC ∩ A λB A λD For homothetic parallelograms there is also a formula for the area.
Statement 1.
Let a quadrangle be convex or re-entrant and its area be S ABCD .The area of a homothetic parallelogram is λ − S ABCD .Proof.
The proof depends on ratio λ . Also, it depends on the type of aquadrangle. It is based on the similatiry of the triangles and summation-subtraction of the areas. Let us prove the statement for a convex quadrangle,with ratio λ < λ = − Yuriy ZakharyanConsidering λ <
0, we obtain: S K λ L λ M λ N λ = S ABCD + S K λ A λB B λA + S L λ B λC C λB + S M λ C λD D λC + S N λ D λA A λD −− S AA λD A λB − S BB λA B λC − S CC λB C λD − S DD λC D λA == S ABCD + (1 − λ ) S OBA + (1 − λ ) S OCB ++ (1 − λ ) S ODC + (1 − λ ) S OAD −− λ S ADB − λ S BAC − λ S CBD − λ S DCA == (cid:0) − λ ) − λ (cid:1) S ABCD = (cid:0) − λ + 2 λ (cid:1) S ABCD == 2( λ − S ABCD (cid:3)
Remark . For the crossed quadrangle
ABCD there is an analogous formula.The area of homothetic parallelogram in this case is 2( λ − S , where S doesnot depend on λ . S is the difference between the areas of the triangles.Taking Corollary 1, Statement 1 and Remark 3, we have: Corollary 3. ∀ λ , λ ∈ R : p λ p λ = | λ − || λ − | , where p λ i means the perimeter of ahomothetic parallelogram with ratio λ i . Here λ can be formally equal to .
3. Perspective
This section is related to the theory of perspective [1, p.70].
Theorem 2.
Homothetic parallelograms are in perspective from the diagonalsintersection point O . (Fig.8) Moreover, ∀ λ , λ ∈ R : (cid:12)(cid:12) OK λ (cid:12)(cid:12) | OK λ | = (cid:12)(cid:12) OL λ (cid:12)(cid:12) | OL λ | = (cid:12)(cid:12) OM λ (cid:12)(cid:12) | OM λ | = (cid:12)(cid:12) ON λ (cid:12)(cid:12) | ON λ | = | λ − || λ − | . Here the denominator can be formally equal to . arignon’s and Wittenbauer’s parallelograms 7Fig.8 Perspective parallelograms Proof.
Let us prove that ∀ λ ∈ R : K λ ∈ OK .It is obvious that K BOA is a parallelogram. Thus, K ∈ OK as it is themidpoint of AB . Also, it is obvious that ∀ λ ∈ R : A λB = B − λA .Thus, the triangles ∆ K λ A λB B λA and ∆ K − λ A − λB B − λA have a common side A λB B λA = A − λB B − λA with the midpoint K .Moreover, other sides being parallel, we have ∆ K λ A λB B λA = ∆ K − λ A − λB B − λA .As a result, K λ A λB K − λ B λA is a parallelogram and K λ , K , K − λ are colin-ear. So, it is sufficient to prove it for the case λ < . Then K , K λ belong toone semiplane from AB . It is obvious that ∆ K λ A λB B λA ∼ ∆ K AB and theirsimilar sides have a common midpoint. Thus, ∠ A λB K λ K = ∠ AK K that means K λ , K , K are colinear. Finally, K λ ∈ OK . For the othervertices the proof is analogous.With the proportionality theorem [3, p.116] we have (cid:12)(cid:12) OK λ (cid:12)(cid:12) | OK λ | = (cid:12)(cid:12)(cid:12) CB λ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) CB λ C (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) CC − λ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) CC − λ B (cid:12)(cid:12)(cid:12) = | − λ || − λ | We should notice that the denominator can be formally equal to zero. Forthe other vertices the proof is analogous. (cid:3)
Yuriy Zakharyan
4. Conclusion
We have seen that Varignon’s and Wittenbauer’s parallelograms are related.Moreover, we can generalize them to homothetic parallelograms (Theorem1). The main properties of the parallelograms can also be generalized (State-ment 1, Corollaries 1–3). It turned out that the homothetic parallelogramsare in perspective from the diagonals intersection point (Theorem 2) and thevertices ratio (the three colinear points ratio [4, p.29]) is predefined (Theorem2).These results were presented at the European Union Contest for Young Sci-entists (EUCYS-2011) [5] and at the International Conference for Young Sci-entists (ICYS-2012) [6].Later in the publication [7, pp.27-36] Romanian mathematician Kiss Sandordefined Wittenbauer-type parallelogram as a special case of the homotheticparallelogram (where λ ∈ Q ∩ (0 , References [1] Coxeter, H. S. M., Greitzer, S. L. (1967).
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Odessa: TypographiyaBlankoizdatelstva M. Shpenzera. (In russian.)[5] Zakharyan, Y. (2011). Research Varignons Theorem, Generalization Witten-bauers and Varignons Theorems, Development of them and use Discoveries inPractice. Presented at the 23rd European Union Contest for Young Scientists,Helsinki, Finland, September 23-28.[6] Zakharyan, Y. (2012). Research Varignons theorem, generalization Witten-bauers and Varignons theorems, development of them. Presented at the 19thInternational Conference of Young Scientists, Nijmegen, Netherlands, April 16-23.[7] Kiss, S. N. (2015). On the Wittenbauer Type Parallelograms.
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