Consider three lines $L_1,L_2,L_3$, with
$L_1 \perp L_2$ at point $A$ and $L_1 \perp L_3$ at point $B$.
$H_A,H_B$ denote point reflections (half turns) at the points $A, B$ respectively.
$R_1$ is a reflection at the line $L_1$ etc.
Prove $H_A R_3 = R_2 H_B$
I found $H_A = R_2 R_1$ and $H_B = R_3 R_1$ so $H_A R_3 = R_2 H_B$ which is $(R_2R_1)R_3 = R_2(R_3R_1)$ by subsitution. Now multiply by $R_2$ so we get: $R_2R_2R_1R_3 = R_2R_2R_3R_1$ and simplify: $I R_1R_3 = I R_3R_1$
I am unsure of where to go now.