Determinants in Elementary Analytic Geometry

Foraker, F. A.

doi:10.2307/2973165

Cited by 2 publications

(4 citation statements)

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“…The exradius is r e = K a−c = 15. By Proposition 8, the excenter I e is = (15,35), and by Proposition 10, the coordinate λ e of the excenter is 10. Similarly, a concave extangential LEQ is shown on the right of Figure 10; its vertices are (0,0), (12,5), (10,5), (6,8), and the side lengths are: 13, 2, 5, 10.…”

Section: Basic Notions For Extangential Leqsmentioning

confidence: 94%

“…One finds there are 979 pairs c, v with v > 4c for which (33) holds. Of these, one finds there is only two where the equation 324 + u 2 = v 2 − (v − 4c) 2 has an integer solution u for which (v + u)/9 is an integer, and such that for the resulting side lengths (a, b, c, d), one has b = min{a, b, c, d}; these are the cases (c, v, u) = (3, 21, 6) and (5, 23, 1), corresponding respectively to the sides (a, b, c, d) = (15,3,3,15) and (37,1,5,41). This completes the proof of the corollary.…”

Section: Proof Of Theorem 1 and Corollarymentioning

confidence: 99%

“…From a standard criteria for a point to be within the circumcircle of a triangle (see [15]), B is inside the circumcircle of the triangle OAC if and only if (74)…”

Section: Lemmata For Extangential Leqsmentioning

confidence: 99%

“…For example, apart from parallelograms and trapezoids, we know of only one convex LEQ that is neither tangential nor extangential. This is the LEQ with vertices (0, 0), (2, 0), (8,8), (8,15) and side lengths 2, 10, 7, 17. There seems to be significantly more tangential LEQs than extangential LEQs, within a ball of any given radius of sufficient size.…”

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: Basic Notions For Extangential Leqsmentioning

confidence: 94%

Section: Proof Of Theorem 1 and Corollarymentioning

confidence: 99%

“…From a standard criteria for a point to be within the circumcircle of a triangle (see [15]), B is inside the circumcircle of the triangle OAC if and only if (74)…”

Section: Lemmata For Extangential Leqsmentioning

confidence: 99%