I am looking for the abstract concept of rotating a point repeatedly.
Equilateral triangle ABC has vertices (0,0), (2,0), (1,$\sqrt{3}$).
If in an equilateral triangle ABC we take point X to be the center of AB at (1,0). We reflect triangle ABC about side BC. What will be the coordinates of point X'(the reflected point X)?
The reflection about side BC makes a new triangle A'BC where A' is the reflected point of A. Now we reflect triangle ABC about side AC. What will be the coordinates of point X''(the reflected point of X')
This is a specific example of my general question: If you wanted to know whether some X'(point X after a given number of reflections) could ever have coordinates (2$\sqrt{3}$, $5\sqrt{3}$) or like (1000,900) after any number of reflections how would you try to find that?
My Progress: I was able to do both of the problems I brought up, but my main concern is the general question.
Thanks!
Suggestion: any point(say $X$) along the line $AB$ is the convex combination of point $A$ and point $B$. So you first find the reflection of $A'$ and do the convex combination of $B$ and $A'$ to get the new point with the same combination coefficient