(Solved): 10. In ABC, suppose the perpendicular bisector of AB is line L1, the perpendicular bisect ...

10. In $?ABC$, suppose the perpendicular bisector of $\overline{AB}$ is line $L_1$, the perpendicular bisector of $\overline{BC}$ is line $L_2$ and the perpendicular 5 bisector of $\overline{AC}$ is line $L_3$. Show that the composition $M_{L_3}?M_{L_2}?M_{L_1}$ gives the single reflection in line $\overleftrightarrow{AO}$ where $O$ is the circumcenter of the triangle. [Hint: Use one of the results we saw in class about compositions of reflections. Find two points fixed by the compositionlook amongst the vertices and $O$.] Bonus: Given three concurrent lines $L_1,L_2$ and $L_3$, construct a triangle $?ABC$ that has the three lines as the perpendicular bisectors of its sides. This means you should give an algorithm for locating the vertices of the triangle in terms of the lines.