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I am willing to teach that the Order of rotational Symmetry of Equilateral triangle as $3$ using Animation. Any suggestions of good applet which demonstrates the rotation of equilateral triangle with different angles like $30^{\circ},60^{\circ},$ etc about the centre?

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    $\begingroup$ If you want, you can easily produce a Desmos diagram showing these rotations (there are ways to animate things in Desmos, and make buttons with prescribed actions). $\endgroup$ Mar 17, 2022 at 11:11

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Here is a simple, dynamic presentation: YouTube video---only 2 minutes, from "Simply Animated Math":

   

For more explanation (and less animation), this may be too detailed, but you could select just the portion on the equilateral triangle: YouTube video (11min total, equilateral triangle: 6min).

   

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    $\begingroup$ (+1) But when something looks wrong (2nd fig.) and then, thinking on it, is wrong, I can't help but being distracted by it. At first, I thought it was an optical illusion brought on by the grid, but then I noticed: Doesn't it bother you that all the vertices and centroid lie on what is apparently a square grid? $\endgroup$
    – user1815
    Mar 20, 2022 at 15:33
  • $\begingroup$ @Raciquel: Sharp eyes! I didn't notice that. $\endgroup$ Mar 20, 2022 at 20:29
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As I mentioned in the comment, things like this are achievable with Desmos. Let me describe a proof of concept here. You can look it up, but please beware that I'm no programmer and no Desmos expert.

If you define a point $A=(1,0)$ and a function $R(P,\alpha)$ that returns the point $P$ rotated by $\alpha$ around $0$, then typing the following action $$ A \to R \left( A,\frac{2\pi}{3} \right) $$ (typed with ->) will produce a button which rotates $A$ by $\frac{2\pi}{3}$. You could make it into an animation by defining $$ B = R(A, \alpha) $$ and adding a parameter $\alpha$. Desmos lets you bound $\alpha$ to the interval $\left[0, \frac{2\pi}{3}\right]$ and animate its changes.

Another method of animating it (I'm not sure if it's better in any way) is to use a ticker that says Run A -> R(A,0.01*dt) every 20 ms. Then the point $A$ will get rotated a bit every 20 ms (dt is the time step, so changing 20 ms to something else won't alter the pace of animation). The problem with this approach is that the rotation doesn't stop after reaching the desired point. In the example I gave, I used a cheap get-around: the animation slows down when it reaches its final point.

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