Let $\triangle ABC$ and straightlines $r$, $s$, and $t$. Considering the set of all mirror images of that triangle across $r$, $s$, and $t$ and its successive images of images across the same straightlines, how can we check whether $\triangle DEF$ is an element of that set?
Given:
- Points: $A(1,1)$, $B(3,1)$, $C(1,2)$, $D(n,m)$, $E(n+1,m)$, $F(n,m+2)$, where $n$ and $m$ are integers numbers.
- Straightlines: $r: x=0$, $s: y=0$ and $t:x+y=5$.
No idea how to begin.
Create an image of your coordinate system, your three lines of reflections, and your original triangle. You can draw in mirror triangles pretty easily, and with a few of these, you will probably find a pattern. (I created my illustration using Cinderella, which comes with a tool to define transformation groups.)
As you can see, there are locations $m,n$ for which the triangle $DEF$ is included. Note that $E$ is the image of $C$ and $F$ is the image of $B$, though. I'll leave it to you as an excercise to find a possible combination of reflections which maps $\triangle ABC$ to $\triangle DEF$.