why I think geometry is HARD

Here are two classic geometry triangle and star polygon proofs that might be really hard at first! Do you think geometry is hard? Especially the geometry problems in the math olympiad or competitions? Well, I certainly think so and I will tell you why!

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[MathAdam]

If you bake a cake in a square pan at 90 degrees, the corners will come out right.

[rigoreyes]

Everyday, he looks more like a wizard. My man is evolving.

[davidcovington]

Clever man. While protesting how hard geometry is, he carries us through 2 beautiful proofs.

[benjamingross]

I teach my students that two of the most powerful tricks in Algebra are adding zero and multiplying by 1. When you complete the square you add zero, and when you add fractions you are multiplying each fraction by 1. In a proof class you have to do this more often and more creatively.

It's similar with geometry. You have to create. But this time instead of creating a clever 1 or 0, you are creating a line or other geometric entity that expands your diagram. It's just a different form of creativity. It comes naturally to some people and less naturally to others.

[disraelidemon]

Thanks for saying this! I watch quite a few geometry videos on YouTube, and I'm always asking myself how you're supposed to know where to start. Most videos are like, "now do this random arbitrary thing that just lets you solve the problem, " as if it's obvious what to do.

[staticchimera]

One of the reasons why geometry is one of my favourite areas of Mathematics is because of the thought that it requires. You have to look at things outside of the box (no pun intended) and you are rewarded for your creativity. Due to this, there are an endless number of possible solutions to just one geometry question.

[gigagrzybiarz]

Speaking about the difficulty of synthetic geometry, I would love you to make a video about using complex numbers in geo problems. I'm sure a lot of people don't know about that topic.

[shrankai]

2:33: The sum of all of the angles of a pentagon is (5-2)*180=540, making the average angle measure to be 540/5=108, with the lines connecting the triangle angles and the pentagon angles making them supplementary angles. Supplementary angles add up to 180, and subtracting the average of 108 from 180, the average angle measure of the non-lettered angle to be 72. 72*2(amount of non-lettered angles)=144, so the average of the lettered angles to be 36. 36*5=180, so there we go!

[agamgujral]

wow... can you do some videos on probability? I love this.

[borissinaga]

Being very used to algebra really make reasoning in geometric terms really hard, at least for me. That's why reading through Euclid's Element is harder for me than reading through calculus books.
What blew my mind was that the ancient greek used geometric reasoning for everything, that's how we get the terms like square of a number, completing the squre and so on. Don't even get me started on Appolonius' Conics, it literally melted my brain.

[blackpenredpen]

Try to prove the “9-point circle” if you want a geometry challenge.

[cmilkau]

3:15 All the triangles you need are already there. The inner angle of the pentagon, say opposing a, forms a triangle with b and e. Hence the corresponding outer angles of the pentagon are b+e. Repeat with outer angles opposing e, obtain a+d. Now these two outer angles form a triangle with c. We conclude a+b+c+d+e = 180° #

[bachirblackers]

I have been waiting for this kind of geo with bprp and i love it .

[soroceandario]

Finally, someone who thinks geometry is hard too...and yes, exactly, you gotta be very observative to know how to do the problems...when I was at 7th grade I loved geometry because it was quite easy...at 8th grade I started to hate it a little bit because it was harder, it required more observation to do it...same thing at 9th grade, I hated it even more, now I finished 10th grade and what i can say, I love algebra so much...and I already watched some of your videos with calculus, which I should study in my next 2 years, and I learned a lot from those videos, thanks to you

[Gustavo_Praga]

I like to prove the star one thinking about the external angles of the pentagon and the 5 triangles. Because actually, the angles a, b, c, d and e are the only ones that aren't from the external angles of the pentagon. So we have that the sum of the 15 angles from the triangles must be 900⁰, but the 10 angles from the bases of the triangles are actually 2 times the external angles of the pentagon, so they are 2×360⁰=720⁰, then we have 720⁰+a+b+c+d+e=900⁰, a+b+c+d+e=180⁰

[littlefermat]

I agree with you sir from my experience as an IMO participant.
Solutions that depend only on Euclidean geometry are really nice but really tough to discover.
How in the world can we know that we need to construct this line, draw this circle, or even define new points?!
Luckily there are other tools that made my life easier (Trigonometry - complex coordinates - barycentric coordinates - projective geometry).
I have discussed these useful tools in short in one of my videos about what topics one should study to prepare for math Olympiad contest.
I also have just started a geometry tutorial on my channel!
So yes Geo is hard, but that's exactly what makes it beautiful 😉

[randomjin]

I'd say that the argument "how in the world do I know to do it" isn't specific to geometry. Just review your own algebra/calculus videos to see you're doing tricks nobody without experience will come up with. Looking at your "my first quintic" or "a brilliant limit"

[neutron]

Fact : Calculus is easier than geometry

[joshuacheung]

To be honest the "star" problem isn't hard... There's a simple solution using exterior angle of triangle where you will obtain a triangle with the angles equal to b, a+d, c+e hence a+b+c+d+e=180 degrees.

[mikefochtman]

When dealing with these sorts of polygons, I always imagine myself traversing along the lines and turning at each vertex. In the case of the triangle, I will have rotated exactly 360 degrees as I traverse. In the case of your star, I would have to rotate a total of 720 degrees (there will be two times that as I rotate around a vertex, I'll be facing 'up' for a moment and I end up facing the same direction I started).

Then, I note how many straight-lines I travel and 'turn off'. Triangle is 3, star is 5. So, the sum of interior angles is <number lines>*180 degrees minus <degrees I rotate> For triangle, 3*180 - 360 = 180. For 'star' you get 5*180 - 720 = 180. For square you get 4*180 - 360 = 360. Hexagon 6*180 - 360 = 720 and so on. Only tricky bit is noting how many times you 'rotate' around as you traverse the diagram.