Lattice equable quadrilaterals. III: Tangential and extangential cases. (English) Zbl 07856171
Summary: 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.
MSC:
| 11D25 | Cubic and quartic Diophantine equations |
| 51M04 | Elementary problems in Euclidean geometries |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
Online Encyclopedia of Integer Sequences:
a(n) = 6*a(n-1) - a(n-2) + 2 with a(0) = 0, a(1) = 3.Pell equation solutions (3*a(n))^2 - 10*b(n)^2 = -1 with b(n) = A097315(n), n >= 0.
Pell equation solutions (7*a(n))^2 - 2*(5*b(n))^2 = -1 with b(n):=A097733(n), n >= 0. Note that D=50=2*5^2 is not squarefree.
Number of representations of n as the difference of two positive squares.
x-values in the solution to the Pell equation x^2 - 74*y^2 = -1.
y-values in the solution to the Pell equation x^2 - 74*y^2 = -1.
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