**SHARPEST ISOSCELES TRIANGLES**

Excel file: 8.SharpestIsocelesTriangles

Chakerian and Erfle October, 2020 paper: 8._SharpestTrianglesPaper

This is a more general and expansive paper than provided for Files 6 and 7. Indeed, the paper for File 6 used the simplest results from this paper in order to create a version accessible to very young students.

This was the first paper I wrote with my mentor, Don Chakerian. This is an early draft of what became the September 2023 **College Mathematics Journal** piece mentioned in File 6. The original impetus for the paper came from some images based on ** n** = 4

**+2 polygons that I shared with Don. I had obtained these images using the second double jump aestheometry File 4.2. (Specifically, you can obtain rotated versions of some the images in File 8 using File 4.2 for**

*k***= 4**

*n***+2 polygons by setting**

*k**=*

**S****= 1 and**

*Jump 1***=**

*P***-1 with last**

*n***jumps counted counterclockwise clicked on in cell H2.) Don was intrigued enough that we started working together.**

*n*As we worked jointly on this paper, Don pointed out to me that there were actually two distinct sets of isosceles triangle images when ** n** is even … my aestheometry files only captured one of them for

*= 4*

**n****+2, and I pointed out to him that when**

*k***is odd, the solution is even easier (but it was also not achievable using my existing aestheometry files).**

*n*This led me to create File 8, but it also led me to create File 9 which allows users to examine all possible triangular images created using the vertices on a polygonal frame.

**P5.** The sections below move beyond isosceles triangles and consider second sharpest triangles in a more general context.