Category: Geometry

I’ve seen variations of this one a few times, so I thought I’d give it a quick write-up. The simpler version is: Given two circles that are tangent and a line that is cotangent to them, what is the length of the segment between the points of tangency? To make things easier, the radii are…

A recent comment from a colleague got me thinking about describing polygons using functions. His intent was that polygons (and all closed shapes) can be described as sets of functions; for instance, a triangle could be described by three linear functions with the domain of the triangle’s vertices. And, of course, any closed shape cannot…

I have borrowed from a colleague a copy of G. A. Wentworth’s Plane and Solid Geometry, copyright 1899 and published 1902 by The Athenæum Press of Boston. I enjoy reading old textbooks because they either reinforce or give lie to certain claims about the longevity of mathematical concepts. This particular volume is attractive to me as…

Today’s lesson in my Geometry class was on the use of the geometric mean when finding missing values of right triangles. For every right triangle, two of its altitudes are the legs and the third is perpendicular to the hypotenuse. The length of the altitude is the geometric mean of the lengths of the two…

In my last post, I noted that it’s possible to create an isosceles trapezoid from four isosceles triangles, but I wasn’t sure if there was a way to construct a quadrilateral from isosceles triangles such that the quadrilateral was neither a rectangle nor an isosceles trapezoid. Now I know that it is not. Let’s reconsider…

We’re working on rigid transformations in my Geometry classes. The basic transformation rules for translation and reflection over a vertical or horizontal line are straightforward; here, they’re written as functions, rather than the briefer vector notation. Translation of \(h\) horizontally and \(k\) vertically: \[(x, y) \rightarrow (x + h, y + k)\] Reflection over a…

Today, I’m going to write up a quick geometric proof. Here’s the original puzzle that inspired it. Given that \(AC = AB\) and that \(AE = DE = CD = BC\), what is the measurement of \(\angle A\)? In other words, the diagram has four isosceles triangles: \(\Delta AED\), \(\Delta EDC\), \(\Delta CBD\), and \(\Delta ABC\). My strategy…

(Edited 6/20/23: I lost the images for this post, and they’re in 3D. I haven’t reconstructed them, so there’s an additional challenge for you!) This is a challenging one: Given all the information at one corner of a tetrahedron (all three surface angles and all three edge lengths), what is the volume of the tetrahedron?…

(Edited 6/20/23: I lost the images for this post, and they’re in 3D. I haven’t reconstructed them, so there’s an additional challenge for you!) [Image lost, not reconstructed.]Here’s a geometry challenge. A plane intersects a cube in such a way as to form a pentagon. If AL, FJ, and CM are all one-fourth of the…

Some time ago, the discussion on π being the incorrect number for calculations in trigonometry, in favor of τ (2π), led me to muse about creating a unit to replace the degree, called the wedge and being equal to nearly two degrees (100W = 60°). I found myself musing about the topic again, but I’ve…