Let $\triangle PQM$ be a triangle. Let $L_1, L_2, L_3$ be the three lines which bisect the three angles of the triangle, respectively. Let $O$ be the point of intersection of $L_1$ and $L_2$. Prove that $O$ lies on $L_3$. [Hint: From $O$, draw the perpendicular segments to the corresponding sides. Prove that their lengths are equal.]
I am currently struggling with this proof and I am honestly lost, the exercise also gives this hint: From $O$, draw the perpendicular segments to the corresponding sides. Prove that their lengths are equal. But I don't see how this could help me, how should I go about solving this?
Since the problem is very standard, I would just give you another hint - the full proof is easily found on internet. Draw a good picture of an angle and its bisector and think about what are the points lying on the bisector of an angle. Can you characterize it without talking about the angle itself? Maybe in terms of distances? After you think about it, try to prove this different characterization - for that the hint in the book is useful.
Also, the hint already suggested you that the distances to the sides of the angle are equal. So the general question is: how would you prove two distances are equal? The basic way (though not super intuitive at first!) of doing that is looking on triangles. Note that if you find two equal triangles, their sides are equal!