# Wilson Stothers' Geometry Pages

What's new?

 june 2002 odds and ends Some thoughts and ideas which do not fit neatly elsewhere. First, a short proof for Lester's Circle, with over a hundred other circles through at least four named centers including the Parry Circle. Second, some thoughts on inversive versions of theorems of Miquel, Ceva and Menelaus. may 2002 a hyperbolic model of euclidean geometry We show how purely hyperbolic concepts can be used to build a model of euclidean geometry in the hyperbolic plane. This has the important consequence that the two geometries are either both logically consistent or both logically inconsistent. more on orthogonal i-lines We present some results on orthogonal i-lines. These have applications in hyperbolic geometry, but also relate to the euclidean theory of coaxial circles. april 2002 asymptotic points and triangles a collection of pages investigating the role of the boundary in hyperbolic geometry further formulae for hyperbolic geometry introducing new formulae related to hyperbolic triangles, and a form of duality which relates lengths and angles. incircles and excircles some formulae for the inradius, and an existence theorem for excircles, including some formulae for the exradius. the latter includes the case of a horocycle as excircle. february-march 2002 hyperbolic polygons During this period, we added a large section (over 30 pages) on hyperbolic geometry. The main object was to prove the existence of hyperbolic polygons with sides of given hyperbolic length. This seems quite tricky, at least if a form of uniqueness result is to be achieved. Along the way, we investigate horocycles, hypercircles, saccheri quadrilaterals. january 2002 hyperbolic circumcircles We discuss an algebraic condition which characterizes the hyperbolic triangles which have a hyperbolic circumcircle. area in hyperbolic geometry We use the gauss-bonnet formula to define the hyperbolic area of a hyperbolic triangle, and deduce some results for hyperbolic polygons. heron's formula in hyperbolic geometry In euclidean geometry, heron's formula gives the area of a triangle in terms of the lengths of its sides. We establish a hyperbolic version. stewart's theorem If AD is a cevian of the euclidean triangle ABC, then Stewart's Theorem gives the length of AD in terms of those of AB, AC,BC,BD and CD. We give a proof, and also establish a hyperbolic version. van obel's theorem in hyperbolic geometry This is the hyperbolic analogue of the euclidean von Obel's Theorem, relating the hyperbolic ratios in which concurrent cevians are cut to other ratios and lengths. december 2001 the hyperbolic circumcircle If A, B and C are points in the disk, there may or may not be a hyperbolic circle through all three. We now investigate the problem of determining, for fixed A,B, the set of C for which the circumcircle exists. This leads to the concept of horocycles. There is also a CabriJava applet to illustrate the result. further theorems on hyperbolic circles Hyperbolic analogues of euclidean theorems, including those on segments of circles. These lead to the analogue of the radical axis of a pair of hyperbolic circles, but now the axis may not exist, even for non-concentric circles. calculations in hyperbolic geometry Some calculations in the Poincare model, such as criteria for points to be collinear in the hyperbolic sense. There are explicit formulae for inversion and equations for a line. proof of pascal's theorem Just that - a proof! von Obel's Theorem We prove a theorem related to Ceva's Theorem. This shows how the ratios in which cevians are cut is related to those of the sides. november 2001 the problem of apollonius Given three objects, each of which may be a point line or circle, the problem is to determine how nay lines or circles can be drawn, touching all three oblects. Since the question involves lines and circles, it is not hard to see that inversive ideas are appropriate. We show that, apart from one cofiguration, which has an infinite family, there are at most eight common tangents. In fact, the real problem is to construct them all. We hope to develop this later. excircles of a hyperbolic triangle A CabriJava applet is used to show that a hyperbolic triangle may have up to three excircles, but may have 0, 1, 2 or 3.