I was playing about with compass and straightedge construction, namely with this site here:

https://sciencevsmagic.net/geo/#

I've made an observation, and I don't understand why it's happening. In my quest to try and understand it myself, I've also found that I can't do basic trigonometry - I'll call that the subproblem.

Subproblem:

Consider a right angle triangle with an unknown hypotenuse, two sides of length 1 and two angles of 45 degrees. Pythagoras' theorem states that the hypotenuse is root 2, about 1.41 . Ok, great, fine.

Now, the sine rules states a/sin(A) = b/sin(B) = c/sin(C). Therefore, 1/sin(45) = hypotenuse/sin(90). So, naturally, I punched in 1/sin(45) and multiplied the answer by sin(90) to get the hypotenuse, which was about *drum roll* ... 1.05. Hang on, two theorems giving two values for the same measurement? Very confused.

Main observation I'm actually trying to understand:

Consider a circle centre A. Points B and C lie on the circumference. Let angle BAC be "A", line BC be "a" and arc BC be "arca".

For any value "n":

The perpendicular to the nth divide of "a" will intersect the nth divide of "A" at a point between "a" and "arca".

More intriguingly, it seems only to work when the nth divide of A is constructible with a compass and straightedge. So, it works when "A" = 90 degrees and n = 3 or 4. It works when "A" = 30 and n = 4, but not n = 3.

Furthermore, let the point of intersect be "D", the nth divide of a be "E" and where the perpendicular to E meets arca be "F". It appears that, when it does work, the ratio of distances ED/DF remains constant regardless of the angle and value for n.

Any insight into why all this is happening (and some help with my appalling trigonometry) would be great.