Coincidences of Centers of Plane Quadrilaterals

Al-Sharif, Abdullah I.; Hajja, Mowaffaq; Krasopoulos, Panagiotis T.

doi:10.1007/s00025-009-0417-6

Cited by 8 publications

(15 citation statements)

References 19 publications

(17 reference statements)

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“…Indeed, one finds readily there are just 20 such pairs c, v with c even and 2v > c for which (30) holds. For only three of these pairs does the equation 16 + u 2 = v 2 − (v − 1 2 c) 2 have an integer solution for u with u + v even; these are (c, v, u) = (4, 2, 6), (2,1,9), (8,4,6), corresponding to the sides (a, b, c, d) = (4, 4, 4, 4), (8,5,2,5), (2,5,8,5) respectively. The last two cases correspond to the same LEQ, up to Euclidean motion.…”

Section: Proof Of Theorem 1 and Corollarymentioning

confidence: 99%

“…One finds there are just 58 pairs c, v with v > c for which (30) holds. Of these, there is only one where the equation 16 + u 2 = v 2 − (v − c) 2 has an integer solution u for which v + u ≡ 0 (mod 3), and such that for the resulting side lengths (a, b, c, d), one has b = min{a, b, c, d}; this is the case (c, v, u) = (4, 7, 2), corresponding to the sides (a, b, c, d) = (5,3,4,6). Assume τ = 5.…”

Section: Proof Of Theorem 1 and Corollarymentioning

confidence: 99%

“…Corollary 3. Up to Euclidean motions, there is only one concave non-kite extangential LEQ; it is the LEQ with vertices (0, 0), (12,5), (10,5), (6,8) and side lengths (13,2,5,10).…”

Section: Introductionmentioning

confidence: 99%

“…Now assume that K A = K C . Then (6) gives (a−d)p 2 = (a−d)(a−b) 2 . Notice that p = ±(a − b) is impossible, as otherwise the triangle OAB would be degenerate.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the above proposition holds in both convex and concave cases, but in the latter case, with a reflex angle at B for example, the signed area K B is negative. For more on the incenter of tangential quadrilaterals, see [6].…”

Section: Introductionmentioning

confidence: 99%

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Lattice Equable Quadrilaterals III: tangential and extangential cases

Aebi¹,

Cairns²

2021

Preprint

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A lattice equable quadrilateral is a quadrilateral in the plane whose vertices lie on the integer lattice and which is equable in the sense that its area equals its perimeter. This paper treats the tangential and extangential cases. We show that up to Euclidean motions, there are only 6 convex tangential lattice equable quadrilaterals, while the concave ones are arranged in 7 infinite families, each being given by a well known diophantine equation of order 2 in 3 variables. On the other hand, apart from the kites, up to Euclidean motions there is only one concave extangential lattice equable quadrilateral, while there are infinitely many convex ones.and suppose the following conditions hold:/s, b = a + c − d and suppose that the above conditions (i) -(iii) hold for t = s s−1 and that b > 0. Then there is a tangential LEQ OABC with successive side lengths a, b, c, d for which (σ, τ ) = (s, t).Furthermore, in both of the above cases, if OABC is concave, then the reflex angle is at B. gives explicit examples: we present calculations of the incenters of LEQs that are kites, and we give an infinite nested family of non-dart concave tangential LEQs. Section 3 gives a series of lemmas on tangential LEQs leading to the definition of the key functions σ and τ , and their properties. In Section 4 we give the proof of Theorem 1 and Corollary 1. Section 5 is the most substantial part of Part 1.Here we prove Theorem 2 and Corollary 2. The final section of Part 1, Section 6, gives more examples. In particular, we show that there are infinitely many LEQs for each of the seven possible choices of (σ, τ ).Part 2 treats extangential LEQs. Sections 7 and 8 follow the general plan adopted in Sections 1 and 3 of Part 1; Section 7 presents some general results for all extangential quadrilaterals, and Section 8 gives a series of lemmas leading to the definition of the functions Σ and T , and their properties. Section 9 treats extangential LEQs in the cases where (Σ, T ) = (9, 18), (18, 50) and (45, 50). Section 10 shows how Theorem 3 can be deduced from Theorem 4. Section 11 is the longest section in the paper; here we prove Theorem 4. This section also contains the proof of Corollary 3, see Remark 31. Finally, in Section 12 we discuss the Open Problem presented above.

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Section: Proof Of Theorem 1 and Corollarymentioning

confidence: 99%

Section: Proof Of Theorem 1 and Corollarymentioning

confidence: 99%

“…Corollary 3. Up to Euclidean motions, there is only one concave non-kite extangential LEQ; it is the LEQ with vertices (0, 0), (12,5), (10,5), (6,8) and side lengths (13,2,5,10).…”

Section: Introductionmentioning

confidence: 99%

“…Now assume that K A = K C . Then (6) gives (a−d)p 2 = (a−d)(a−b) 2 . Notice that p = ±(a − b) is impossible, as otherwise the triangle OAB would be degenerate.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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