#Need help understanding euclidean

126 messages · Page 1 of 1 (latest)

pseudo mantle
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techie pls help 🙏

muted coralBOT
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cunning dagger
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So the first thing to understand is that Cartesian coordinate systems don't change anything about geometry. They just let us talk about geometry more easily.

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They let us describe points with ordered pairs and lines and circles with equations, but they don't change what points, lines, and circles are or how they behave.

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Consider Euclid's first proposition, constructing an equilateral triangle from a line segment.

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Let's consider the line segment AB in the Cartesian plane. Arbitrarily, let A = (0, 0) and B = (l, 0).

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l is therefore the length of line segment AB.

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Now let us consider the circle centered at A with radius AB.

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In the Cartesian system, that circle is described by the equation x^2 + y^2 = l^2.

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The circle centered at B with radius AB is (x - l)^2 + y^2 = l^2.

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Now let's find the intersection of these circles.

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x^2 + y^2 = (x - l)^2 + y^2
x^2 = x^2 - 2lx + l^2
2lx = l^2
x = l/2

y = +/- sqrt(l^2 - (l/2)^2)
y = +/- sqrt(3)/2 l```
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Thus the intersections are (l/2, sqrt(3)/2 l) and (l/2, -sqrt(3)/2 l).

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Arbitrarily choose (l/2, sqrt(3)/2 l) and call it C.

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Now we know length AC is l, because C is on the circle centered on A with radius l. And we know length BC is l because C is on the circle centered on B with radius l.

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We can also just calculate the length directly with the distance formula.

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Which is just the Pythagorean theorem, which we do therefore need to prove separately, but that's relatively simple.

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Also the distance formula is where we get the equation for a circle.

cunning dagger
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@pseudo mantle

pseudo mantle
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but how can I use it to prove certain theorems and get certain angles?

cunning dagger
pseudo mantle
cunning dagger
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There necessarily exists exactly one circle with A, B, and C on its circumference.

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So just draw that circle and see if D's on it.

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What a Cartesian coordinate system adds to the process is the ability to describe all the points and lines and circles with algebraic precision.

pseudo mantle
# cunning dagger So just draw that circle and see if D's on it.

ahhh I like how u make me think about it from a different perspective
is there any other tricks to solve for certain angles? like I know for a cyclic quad the exterior angle is equal to the interior opposite
however, certain things for euclidean like finding angles are not easy for me

cunning dagger
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So you don't have to eyeball whether D is on the circle, you can just plug D's coordinates into the equation.

pseudo mantle
pseudo mantle
cunning dagger
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The big benefit of Cartesian coordinates is the ability to convert back and forth between algebra and geometry at will. To demonstrate why this is, tell me, how would you find the circumcircle of a triangle?

pseudo mantle
cunning dagger
pseudo mantle
cunning dagger
pseudo mantle
wicked raven
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bruh

cunning dagger
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The circle which passes through all of its vertices.

pseudo mantle
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ohhhhhhhhhhhh

cunning dagger
pseudo mantle
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yeah and then u can find angles in a semi circle = 90 degrees right?

cunning dagger
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...what?

pseudo mantle
# cunning dagger ...what?

what I've been taught, is when you see the triangle in the circle then its angles in the semi circle
but that doesnt look like its cutting it correctly to become an angle in the semi circle

cunning dagger
pseudo mantle
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I suck at geometry more than anything, its weighing me down so all I know is this and that

pseudo mantle
cunning dagger
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The article is not what makes it not a sentence.

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I'm sorry, but do I seriously have to explain to you what does or doesn't make something a sentence right now?

pseudo mantle
pseudo mantle
cunning dagger
pseudo mantle
cunning dagger
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I guess I am going to have to explain how sentences work.

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A sentence has three parts. Subject, verb, object.

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"Its angles in a semicircle" is a noun phrase. It's only a subject or an object.

pseudo mantle
pseudo mantle
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"it's angles in a semi circle", thats what I meant

cunning dagger
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Okay, what does that mean?

pseudo mantle
cunning dagger
pseudo mantle
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idk how to explain, lemme get a picture or smth

cunning dagger
cunning dagger
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Not to be mean.

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But because you can't learn until you admit you don't know.

pseudo mantle
pseudo mantle
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ask me anything in algebra I can use my head and try to figure it out
but, for geometry I'm like a dumbass

cunning dagger
pseudo mantle
cunning dagger
pseudo mantle
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anything please, before the end of this year I needa master euclidean coz I need that report to be juicy with those good marks for math

cunning dagger
pseudo mantle
cunning dagger
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No, dude.

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You look ignorant because you are ignorant. You're here because you're ignorant, to become less ignorant. What makes you look like an idiot is trying to pretend you're not ignorant.

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Here's how it's gonna go. I'm gonna explain the thing you think you understand. If you remember something you feel is related, keep your mouth shut. If something seems logical, then you can bring it up.

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Deal @pseudo mantle?

pseudo mantle
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ok go ahead

cunning dagger
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Okay.

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So first let's talk about circumcircles.

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Every triangle has a circumcircle, and we're gonna prove it.

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A circumcircle is a circle which passes through all the vertices of a polygon.

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A polygon is a shape made of line segments that share endpoints, and its vertices are those endpoints.

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So what does it mean for a triangle to have a circumcircle? What is a circle?

pseudo mantle
cunning dagger
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No.

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A circle is the set of all points equidistant from a center point.

pseudo mantle
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ahhh wait visually thinking about it, that makes sense

cunning dagger
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Right.

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Which means, if the vertices of a triangle, call them A, B, and C, are all on the same circle, maybe call its center O, what does that imply?

pseudo mantle
cunning dagger
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I mean, we have these premises:```

  1. A, B, and C are all on the same circle with center O.
  2. A point is on a circle with radius r and center X if and only if it is a distance r from X.```What conclusion can we draw from this?
pseudo mantle
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call me stupid behind the screen but I'm poor at this

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but I want to learn

cunning dagger
pseudo mantle
cunning dagger
pseudo mantle
cunning dagger
pseudo mantle
cunning dagger
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...did I say the word "tangent"?

cunning dagger
cunning dagger
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@pseudo mantle The conclusion is "A, B, and C are all a distance r from O."

pseudo mantle
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why doesnt the esc key work anymore

cunning dagger
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It didn't work for me a few days ago but does now.

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Anyway.

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So we wish to prove that there exists a point which is the same distance from A, B, and C.

pseudo mantle
cunning dagger
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The point is arbitrary because A, B, and C are arbitrary (but noncolinear).

pseudo mantle
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ok I'mma go sleep, I'll reply when I'm awake

cunning dagger
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Fair enough.

inland mauveBOT
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@pseudo mantle

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