Circles and Triangles (And the Snowflake which Results)

Today we discuss circles and triangles, particularly inscribed circles and triangles.

For the past few days at work, while I’ve been waiting for the internet to load, I’ve been working on a problem.

A few weeks ago, my boss made me clean the supply cabinet and I found a very nice compass set. When I asked my boss about it, and told her we could probably have a garage sale and someone would love to buy it, she said, “Those things have been marked out of stock, and we cannot sell them.”

“What!” I thought to myself. “That’s so wasteful! It's going to sit in this cabinet forever and no one will ever use it or even know it's there!”

Anyway, this is not about unethical work habits, but MATH (which in this case made me cry because it was sitting unused in the supply cabinet).

So, when I found this compass, along with many other things which are completely useless in our workplace, I took it to my desk and called it the Shipping Manager’s (mine).

I may or may not be obsessed with snowflakes, so as I sit around, waiting for the internet to load, I like to amuse myself by drawing snowflakes. I like to make my snowflakes realistic, so when I draw or cut out snowflakes, I make them six sided. This is a little harder than the typical eight sided when cutting, and quite a bit harder when drawing. This compass however, proves miraculous when making equilateral triangles.

For those of you who don’t know how to construct an equilateral triangle with a compass, it is simple. Create a circle. Create another circle, using any point of the first circle as the center, and using the same radius as the first circle. Like so:

By the way, the software I use for geometry is called WinGeom, and is a free software I downloaded from the internet. It is similar to Geometer’s Sketchpad, but free (though every time I see Geometer’s Sketchpad lying on the shelves at work I think about buying it).

You may then proceed to draw another equilateral triangle of the correct size, and you get something that looks like the Star of David, and you may proceed by chiseling out whatever holes you like in your snowflake. In my love of snowflakes, I already see so many ways to begin just from the work we did to construct such a beautifully perfect set of triangles. 

So, now to the work problem.

I was sitting in my desk, looking at my compass and thinking about snowflakes, and how the two circles use so much space for such a small triangle. I wanted something better. I decided to make an equilateral triangle inscribed in a circle. This would give me more creative ability when I got to the snowflake part as I would have more room on the paper.

And I came up with the drawing below. Begin with a circle. Chose a point on the circle, C. Draw ray AC. Bisect the segment AC. Draw circle AE. Mark point G. Draw tangent GH. Mark points IJ. Make Triangle! and then you have an inscribed circle too!

Needless to say, today was about the most amazing day of my life even though I didn’t figure this out until I got home and did math with dry erase markers on my windows. (I used some trigonometry to discover this method. I began with a poorly drawn circle and a poorly drawn equilateral inscribed triangle. If you draw segments from the center to the vertices you get three isosceles triangles. Each triangle has two 30 degree angles and a 120 degree angle. If you divide these triangles in half, you get six 30-60-90 triangles which have something in common with the unit circle, and then sine and cosine and then TADA!*)

Then, I sat down to my computer and started constructing said triangles and circles.

Now, consider the pink circle (I have a close up below). In this circle, F0 bisects segment XD0, H0 bisects F0X, and G0 bisects D0F0.

In this pink circle is green triangle I0C0B0. Point Z slides along ray YD0.

Now, I knew that a triangle formed with the diameter of the circle and any point on the circle will produce a right angle at the point on the circle no matter where the point on the circle is. I wondered: if I have a chord C0B0 and move it to some random location in the circle, and then add point I0 to make a triangle, if I move I0 along the circle, will the angle at I0 be constant? Yes! This added even more excitement to my day. 

(This is the beauty of geometrical software. It allows you to see if a hypothesis is true before you do all the work to prove it is true only to find it was false.)

Now I just have to prove why…

*My purpose is not to make you cry, so please leave me a message if you would like further explanations of my process. 
Also, I tend to do things the hardest way possible. Be warned.