Lattice Equable Quadrilaterals III: tangential and extangential cases

Abstract

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.