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Introduction

## Publications

Publications (74)

This is a preprint of work in progress describing methods of constructing ellipses and elements related to ellipses. Please let me know if I am missing any important constructions.

We study some properties of a triad of circles associated with a triangle. Each circle is inside the triangle, tangent to two sides of the triangle, and externally tangent to the circle on the third side as diameter. In particular, we find a nice relation involving the radii of the inner and outer Apollonius circles of the three circles in the tria...

Two ellipses, one outside the other, divide the plane into three regions. Let T be the common self-polar triangle of the two ellipses. We show that each of the three vertices of T lies in a different region. The polar of a point with respect to a conic and the pole of a line with respect to a conic are well known concepts in projective geoemtry. Se...

We study properties of certain circles associated with a triangle. Each circle is inside the triangle, tangent to two sides of the triangle, and externally tangent to the arc of a circle erected internally on the third side.

Let P be a point inside a convex quadrilateral ABCD. The lines from P to the vertices of the quadrilateral divide the quadrilateral into four triangles. If we locate a triangle center in each of these triangles, the four triangle centers form another quadrilateral called a central quadrilateral. For each of various shaped quadrilaterals, and each o...

The diagonals of a quadrilateral form four component triangles (in two ways). For each of various shaped quadrilaterals, we examine 1000 triangle centers located in these four component triangles. Using a computer, we determine when the four centers form a special quadrilateral, such as a rhombus or a cyclic quadrilateral. A typical result is the f...

The first isodynamic point of a triangle is one of many notable points associated with a triangle. It is named X(15) in the Encyclopedia of Triangle Centers. This paper surveys known results about this point and gives additional properties that were discovered by computer.

A cevian of a triangle is a line segment that extends from a vertex of the triangle to a point on the opposite side. A cevian that passes through a triangle center is called a central cevian. There are a number of inequalities known concerning central cevians. For example, if m a , m b , and m c are the lengths of the medians of a triangle, then it...

The incircle of a triangle touches the sides of the triangle in three points. It is well known that the lines from these points to the opposite vertices meet at a point known as the Gergonne point of the triangle. We use a computer to discover and catalog properties of the Gergonne point.

Using a computer, we find the formula for the distance between a number of triangle centers in terms of R, r, and s (the circumradius, inradius, and semiperimeter of a triangle). We then use the fact that this expression is always nonnegative to find upper and lower bounds for powers of the variables R, r, and s in terms of the other variables.

Let P be a point inside a convex quadrilateral ABCD. The lines from P to the vertices of the quadrilateral divide the quadrilateral into four triangles. If we locate a triangle center in each of these triangles, the four triangle centers form another quadrilateral called a central quadrilateral. For each of various shaped quadrilaterals, and each o...

We use a computer to determine the relative location of Kimberling centers X(1) through X(100) with respect to the incircle of a triangle. For example, for all triangles, we determine which centers must lie inside the incircle and which must lie outside the incircle. We also present inequalities representing how far away these centers can get from...

The first isodynamic point of a triangle is one of many notable points associated with a triangle. It is named X(15) in the Encyclopedia of Triangle Centers. This paper surveys known results about this point and gives additional properties that were discovered by computer.

A cevian is a line segment joining the vertex of a triangle and a point on the opposite side. Well-known cevians are medians, angle bisectors, and altitudes. We consider various cevians passing through named triangle centers such as the Gergonne point and Nagel point. We show how a computer can be used to discover, not just prove, identities involv...

A Gergonne cevian is the cevian through the Gergonne point of a triangle. A Nagel cevian is the cevian through the Nagel point of a triangle. We present some new inequalities involving the lengths of the Gergonne and Nagel cevians of a triangle. Mathematica was used to both discover and prove some of these results.

If P is a point inside triangle ABC, then the cevians through P extended to the circumcircle of triangle ABC create a figure containing a number of curvilinear triangles. Each curvilinear triangle is bounded by an arc of the circumcircle and two line segments lying along the sides or cevians of the original triangle. We give theorems about the rela...

We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of the tetrahedron. Using a computer, we determine when any of 16 conditions occur (such as the four centers bei...

We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of the tetrahedron. Using a computer, we determine when any of 16 conditions occur (such as the four centers bei...

If P is a point inside ABC, then the cevians through P divide ABC into six small triangles. We give theorems about the relationships between the areas of these triangles.

If P is a point inside triangle ABC, then the cevians through P extended to the circumcircle of triangle ABC create a figure containing a number of curvilinear triangles. Each curvilinear triangle is bounded by an arc of the circumcircle and two line segments lying along the sides or cevians of the original triangle. We give theorems about the rela...

If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into smaller triangles of various sizes. We give theorems about the relationships between the radii of various circles associated with these triangles.

If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into smaller triangles of various sizes. We give theorems about the relationships between the radii of various circles associated with these triangles.

If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into six small triangles. We give theorems about the relationships between the radii of the circumcircles of these triangles. We also state some results about the relationships between the circumcenters of these triangles.

If P is a point inside triangle ABC, then the cevians through P extended to the circumcircle of triangle ABC create a figure containing a number of curvilinear triangles. Each curvilinear triangle is bounded by an arc of the circumcircle and two line segments lying along the sides or cevians of the original triangle. We give theorems about the rela...

If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into six small triangles. We give theorems about the relationship between the radii of the circles inscribed in these triangles and the lengths of the segments formed along the sides of the triangle.

If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into smaller triangles of various sizes. We give theorems about the relationship between the radii of certain excircles of some of these triangles.

If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into small triangles. We give theorems about the relationship between the radii of the circumcircles of these triangles.

If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into six smaller triangles. We give theorems about the relationship between the radii of the circles inscribed in these triangles.

If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into six smaller triangles. We give theorems about the relationship between the radii of the circles inscribed in these triangles.

Let P be a point inside triangle ABC. The cevians through P divide triangle ABC into 6 smaller triangles.
Consider the incircles of these 6 triangles. I am investigating the relationship between the radii of the 6 incircles.
There must be some relationship between these six radii, but I have not found a general formula yet. Here are some particul...

If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into smaller triangles of various sizes. We give theorems about the relationship between the radii of certain excircles of some of these triangles.

If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into six small triangles. We give theorems about the relationship between the radii of the circles inscribed in these triangles and the lengths of the segments formed along the sides of the triangle.

Paper 13: Stanley Rabinowitz and Stan Wagon, “A spigot algorithm for the digits of π,” American Mathematical Monthly, vol. 102 (March 1995), p. 195–203. Copyright 1995 Mathematical Association of America. All Rights Reserved.
Synopsis: In this paper, Rabinowitz and Wagon introduce a very interesting “spigot algorithm” for the digits of π. In partic...

In one general aspect, a data access method is disclosed that includes directing data block write requests from different clients to different data storage servers based on a map. Data blocks referenced in the data block write requests are stored in the data storage servers. Data from the data write requests are also relayed to a parity server, and...

Am odel is described and analyzed for a multiprocessor shared memory system in which each memory bank can service a fixed number of access requests per cpu cycle. If n processors simultaneously request data from a common shared memory, it is usually not possible for all the requests to be satisfied at the same time. This is because the memory syste...

Some Bonnesen-style isoperimetric inequalities for triangles in the plane are presented. For example, it is shown that L 2 -12√3A ≥ 35.098 r(R -2r) for triangles with perimeter L, area A, inradius r, and circumradius R. Equality holds when and only when either the triangle is equilateral or the triangle is similar to the isosceles triangle with sid...

Mathematical journal and contest problems from 1975-1979 arranged by subject. Includes author index.

Algorithms for evaluating or simplifying sums of the form $$\sum_{n=1}^{N}\frac{1}{F_{n+a_1}F_{n+a_2} \cdots F_{n+a_r}}$$ where the F
i are Fibonacci numbers and the a
i, are integers have been discussed in [13]. It is the goal of this paper to generalize these results to arbitrary second-order linear recurrences.

Methods for manipulating trigonometric expressions, such as changing sums to products, changing products to sums, expanding functions of multiple angles, etc., are well-known [1], In fact, the process of verifying trigonometric identities is algorithmic (see [2] or [5]). Roughly speaking, all trigonometric identities can be derived from the basic i...

This paper deals with linear systems of difference equations whose coefficients admit generalized factorial series representations at z = ∞. We shall give a criterion by which a given system is determined to have a regular singularity.
In the same ...

Given a convex lattice polygon with g interior lattice points, we find upper and lower bounds for the perimeter, diameter, and width of the polygon. For small g, the extremal figures were found by computer. A lattice point in the plane is a point with integer coordinates. The set of all lattice points in the plane is denoted by Z2 .A lattice polygo...

Let S b ea set of mn +1 lattice points in En. Then either some two points of S span a hole (have a lattice point not in S between them), or some m +1 points of S are collinear.