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5h
comment Circumference of hyperbolic circle is $2\pi \sinh r$
Do you want an integral proof? Using the Poincaré model metric tensor $ds = \frac{2 \sqrt{dx^2+dy^2}}{1-x^2-y^2}$, and parameterizing the origin-centered circle with Euclidean radius $R$ by $x = R\cos t$, $y = R\sin t$, the hyperbolic circumference of the circle is $$\int_{0}^{2\pi}\frac{2 R dt}{1-R^2}=\frac{4\pi R}{1-R^2} \qquad(\star)$$ Since Euclidean distance $R$ from the origin corresponds to hyperbolic distance $r=\log\frac{1+R}{1-R}$, we have $R = \frac{\exp r - 1}{\exp r + 1}$. Substituting into $(\star)$ gives $2\pi\sinh r$.
5h
comment Are circles and lines in two-space one-dimensional?
Related: "Why the unit circle in $R^2$ has one dimension?", "Confused about dimension of circle", "Why is a circle one-dimensional?".
10h
revised Circumference of hyperbolic circle is $2\pi \sinh r$
added 9 characters in body; edited title
10h
comment Circumference of hyperbolic circle is $2\pi \sinh r$
I don't see how the model matters. Here's a proof that doesn't reference a particular model.
21h
comment Where can I find a good drawing software?
Satimage's "Smile" leverages the Mac's AppleScript language to draw images programmatically. The interface is poor, and the documentation difficult to navigate, but Smile has some built-in geometric primitives and labeling options (including TeX support) that could be helpful. I've used it to create diagrams such as those shown in this answer. See also some of Satimage's sample images‌​.
21h
awarded  Electorate
1d
comment An equation involving ratios in a triangle.
Good. Write up your solution as an answer, and I'll up-vote it. :)
1d
comment An equation involving ratios in a triangle.
Multiplying the numerator and denominator of the right-hand side by $r/2$ (where $r$ is the inradius), the fraction becomes $|\triangle ABC|/|\triangle IBC|$, which reduces to the ratio of the altitudes of those triangles. The altitudes make convenient similar triangles with $\overline{AD}$ and $\overline{ID}$.
1d
comment What is reflection across parabola?
Note that parabolas of the form $y=h x^2$ and ellipses of the form $x^2 + 2 y^2 = 2 k^2$ comprise an orthogonal family, with which we can assign unique $hk$-coordinates to every point in the plane (taking $k$ negative in the left half-plane). If there's any justice, these families are preserved under parabolic reflection: ellipses invariant (as point-sets), and parabolas permuted. That is to say, reflection takes $(h,k)$ to some $(h^\prime, k)$. The question then becomes: How are $h$ and $h^\prime$ related? (Surely, $h=1$ iff $h^\prime=1$, and $h=\pm \infty$ iff $h^\prime=\mp \infty$.)
1d
revised I think I see mysterious lines inside triangles—how to prove their existence?
added 26 characters in body
1d
comment I think I see mysterious lines inside triangles—how to prove their existence?
@Alexey: I would've beaten you if I didn't take so long editing. ;) I should probably just delete this answer. (Well ... hmmm ... someone just up-voted me. I guess there's perceived value in this near-duplicate, so I'll leave it. :)
1d
revised I think I see mysterious lines inside triangles—how to prove their existence?
added 59 characters in body
1d
answered I think I see mysterious lines inside triangles—how to prove their existence?
2d
comment What is the name of this geometric shape?
I don't believe there's an official name for this shape, so the trick is to find a concise descriptor. "Concave bisymmetric hexagon" would (I think) cover #1, but there's no elegant counterpart for #2. Of course, you're free to call these things whatever you like. I might be inclined to play off of the resemblance to an anvil.
2d
answered Is there another way to solve this Trigo in series?
May
21
comment How to prove that a straight line is an infinite set of points?
When you fill-in the "...", I'll remove my comment. :)
May
21
comment How to prove that a straight line is an infinite set of points?
See my comment under @lixu's answer.
May
21
comment How to prove that a straight line is an infinite set of points?
Of course, this assumes a correspondence between geometric lines and algebraic linear equations of real-number variables. There are geometries with no such correspondence. As mentioned by myself and others, one must be clear about the axioms system being used.
May
21
comment How to prove that a straight line is an infinite set of points?
Importantly, you need that the axiom that at most one of $[A,B,P]$, $[A,P,B]$, and $[P,A,B]$ hold for a particular trio of points $A$, $B$, $C$. (Otherwise, you get "circularity", and your argument doesn't necessarily hold. See my comment under Jaood's answer.)
May
21
comment How to prove that a straight line is an infinite set of points?
"... and you can keep going on like that." Whether this leads to an infinitude of points depends on your axioms. My universe might consist of exactly three points, $A$, $B$, $C$, where I declare each one to be "between" (or even "the midpoint of") the other two. In this case, your construction does not provide infinitely-many points; it just cycles through the set $\{A, B, C\}$ over and over again.