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.
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.
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.
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
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.
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
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.
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.
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.