So I’ve been doing some thinking, and realized that it is about time that I post a blog specifically about math, or specifically giving you some math knowledge. Be prepared to learn (hopefully).
In case it wasn’t made clear, I find math to be beautiful in the way that it works perfectly, in the way that it serves to show patterns, and how it can be used to prove all sorts of different concepts. Therefore, it would make sense that I get a lot of happiness from proofs, specifically the visual proofs of geometry. Yes, I’m a bit of a geometry nerd.
A friend of mine recently taught a lesson in my college geometry class. I was so impressed with the way that she presented the “Nine-Point Circle” proof and was inspired to write! The Nine-Point Circle proof is absolutely beautiful because it shows that you can make a nine sided polygon within a circle – all nine vertices fitting within the same circumference!
Big deal you say? We’ll yes, if you don’t realize that as a polygon increases in the sides that it has, the probability of fitting it into a circle, where all the vertices touch, diminishes (unless of course it is a regular polygon…). Therefore, nine vertices in a non-regular polygon, is sorta a big deal! Sorry for the abundance of mathematics vocab that just got thrown out at you – that can be one of the most challenging parts of following a proof, so thanks for staying with me here.
Anyway, that proof can be slightly complex if you don’t have the foundational knowledge to back it up, so I wouldn’t spew it all out at you. Instead, I wanted to show you that any triangle can have a circle drawn around it. The triangle is the only polygon that, in ALL cases, can be inscribed (fits perfectly inside) a circle where all the vertices (the points of the triangle) touch the circumference of the circle. So lets begin!
First, we say what we are going to start with. Lets start with a triangle, and label the points A, B, and C. Simple.
So we say, "Let ABC be a triangle" and then, for ease, provide a picture.
Next, we state our goal, what do we want? We want to Show that a circle passes through the points A,B and C.
That wasn’t hard, and now we get into applying some prior knowledge about math. Our next step is to make some perpendicular bisectors. Perpendicular bisectors divide line segments in half AND they make right angles (90˙angles), so they are pretty cool.
Lets make two perpendicular bisectors. One to divide line segment BC and one to divide line segment CA. We'll call the points where they divide the lines, m and n respectively, and we can call their intersection point o.
Therefore, because they were perpendicular bisectors, they divided the lines into two equal parts. So we know BM=MC and CN=NA. There is a property that everything is equal to itself called the Reflective property, so OM=OM.
Basically we can say segments OM=OM and segments ON=ON. We can also draw a line to connect B to O and C to O. So now we have two mini triangles. ∆BOM an ∆MOC.
These triangles are congruent because of a math fact from high school geometry, does anyone remember SAS? Side-angle-side. If it can be shown that two triangles have a congruent set of sides, angles and sides (in that order) then the triangles are congruent. Here we’ve seen that these triangles have two sides equal (BM=MC, and OM=OM) and we know that the perpendicular bisector created a right angle, that they both share, so they have the angle equal too. Therefore, ∆BOM = ∆MOC. That was a lot of information right there. 
Because these triangles are equal, we know BO=OC. So if we had a circle with the center at point O, BO and OC could act as radii to the points on the circle which include the vertices B and C.
The next step is repeating all of that for AO so we can show that AO=BO=CO, but instead of repeating we can say “Similarly, AO=CO.” At this point we’re done. Once these lines are proven equal we have three radii that meet at the vertices and it is proof that there is a circle around ∆ABC, which can be ANY triangle.
Admittedly, I didn’t present that in an exciting manner, and I’m not quite sure if many continued reading this far. If you did, just reflect on how, starting with just one triangle, we were able to prove, with undoubtable evidence, that any triangle can be inscribed into a circle. Isn’t that beautiful!?
It is beautiful! It seems obvious that anyone can just “draw” a circle around a triangle–but that doesn’t mean really anything unless you really understand the triangles and circles are more than just shapes–they’re things with real properties.
I feel honored that I am discretely included in this post. I am glad you enjoyed my lesson and were inspired to write about it 🙂 Your blog is amazing… did you know I’ve been reading it?
And yes… math is beautiful!!!!!!!
I just saw this Mara!
Aww, this means so much to me, even if you only have read a few- thank you!