Angle Trisection: Learn a New Solution Here!

[otagotasolo]

Hello everyone,

i stumbled upon a trisection solution that was unknown to me and looked incredibly great (i am not an expert at all)

http://andreasaronsson.com/guides/perspective-drawing/divide-into-equal-parts/

could someone please validate it and explain why it works and why it is not broadly known?

 

[micromass]

The constructions in the website are not angle trisection. They trisect a square, or equivalently, a line segment. This is pretty well known.

 

[otagotasolo]

but one might put the square on the bottom of the angle where angle's bisection line comes in the middle of the square.

 

[micromass]

I don't know what you mean. Angle trisection is impossible with ruler and compass anyway (but possible with origami).

 

[otagotasolo]

i mean that if there is an angle, one might construct an square between the lines of the angle and split this square into 3 parts. wouldn't be the lines trisecting the area trisect also the angle?

 

[micromass]

i mean that if there is an angle, one might construct an square between the lines of the angle and split this square into 3 parts. wouldn't be the lines trisecting the area trisect also the angle?

I don't know what 'square between the lines of the angle' means.

 

[otagotasolo]

ok, sorry for my English. Imagine capital A letter, the dash in the A would be the upper edge of the square

 

[micromass]

OK. I got it.

No, this does not yield a trisection of the angle.

 

[otagotasolo]

thats the part i don't understand exactly. if i split the area into 3 parts, why the points where the lines touch the edge are not splitting the angle.
is it because the angle connection between 2 upper contact points (between the lines and the square) would be round and the square is a straight line?
or is it something else?

 

[HallsofIvy]

That is a very old mistake. Crossing an angle with a line perpendicular to the bisector of the angle, trisecting the line segment, then drawing lines from the vertex through the line segment trisecting points does NOT trisect the angle.

 

[micromass]

thats the part i don't understand exactly. if i split the area into 3 parts, why the points where the lines touch the edge are not splitting the angle.
is it because the angle connection between 2 upper contact points (between the lines and the square) would be round and the square is a straight line?
or is it something else?

Try to prove that it does trisect the angle. Then you will see your error.

 

[micromass]

But to give you an idea. Try to draw an angle very very close to 180°
Then trisect that angle using your method. You'll see straight away that it doesn't work.

 

[otagotasolo]

i don't think i am qualified for that. that's why i asked what is wrong about believing it might trisect the angle

 

[otagotasolo]

ok i will try it

 

[micromass]

i don't think i am qualified for that. that's why i asked what is wrong about believing it might trisect the angle

You have to appreciate the fact that it is impossible to answer this question. You have a certain belief that is not based on rational arguments but merely on "this should work". This is not a bad thing, everybody has these kind of ideas and some of them work brilliantly. This is how ideas in science originate. The point in math is then to back up this intuitive idea with a rigorous proof. This is where you'll find that your idea is either flawed, or it works.

Sure, I can see there is merit in your ideas. And at first, I might agree with you that it might work. But since you have presented no rational and rigorous argument for why it works, it is impossible for me to say why your idea does not work. There simply isn't enough substance here. The only thing we can do is try a proof that it does or does not work. After this, we find that it doesn't. The sensible thing is to abandon the idea, even though it still seems intuitively appealing.

"Why doesn't the sun revolve around the earth?", "Why doesn't light come from my eyes when I see objects". Those are all very decent hypotheses. They just don't work. Why not? Because they don't.

I'm sorry if this is not the answer you expect.

Also, try to download and use a program called geogebra. It's very useful to make these kind of constructions with good accuracy.

 

[otagotasolo]

it looks like drawn in the perspective so it "feels good" but the broader the angle the less equal the parts get.

 

[micromass]

it looks like drawn in the perspective so it "feels good" but the broader the angle the less equal the parts get.

Exactly. To trisect an angle, you need to trisect the circle segment "build inside the angle", and not the line segment. The two are not equivalent.

 

[micromass]

Do you know trigonometry?

 

[otagotasolo]

regarding your 'long' message: I assumed that this is not working because otherwise it would be a known solution. but it felt really good (i used that for drawing, not for calculations)

 

[micromass]

regarding your 'long' message: I assumed that this is not working because otherwise it would be a known solution. but it felt really good (i used that for drawing, not for calculations)

I will agree it feels really really good. It does so with me too. But closer inspection reveals that it doesn't work.

 

[otagotasolo]

Do you know trigonometry?

didnt use in like for a decade but yes.

talking of which: does this method work work well for the angles in the area where the tangent function looks quite like a line and stops working when it starts bending upwards and then go to infinity?
(i mean the 90* which is half of the angle we discussed but there is a symetry involved)

 

[jedishrfu]

As an aside, trisection can be solved using Origami:

https://plus.maths.org/content/trisecting-angle-origami