Given three lines $l,m,n$ with one intersection point $O$, and one point $A$ on line $m$. Prove that there exist a triangle which has three angle bisectors $l,m,n$ and one vertex $A$. I tried to draw a circle with center $O$, which should be tangent to the three sides, But I can not prove it, namely the circle should be tangent with the opposite side $BC$.
Accurate Original Wording :
Basically I am giving as an answer what Calvin Lin put into the comments. But there are some subtle points related to the possibility that the intended incenter could turn into an excenter!
Let $B$ lie on $l$ and $C$ lie on $n$. Then side $\overline{BC}$ makes angle measuring $\frac12\angle B$ with $l$ and $\frac12\angle C$ with $n$. From the angle chasing described by Calvin, you find that in $\triangle OBC$ the vertalex angle at $O$ measures $90°+\frac12\angle A$.
So to get the half-angle at $A$, subtract a right angle from the angle between $l$ and $n$ in the quadrant opposite $A$. Then:
$B$ lies on $l$ such that $\angle OAB$ measures the required half-angle and line $n$ passes between $A$ and $B$ (this betweenness criterion is necessary for $O$ to be an incenter).
$C$ lies on $n$ such that $\angle OAC$ measures the required half-angle and line $l$ passes between $A$ and $B$.
Now comes the subtle part, which (to my eyes anyway) is suggested by the diagram in the original question. $\angle BOC$, defined above from lines $l$ and $n$, could measure less than $90°$! When this happens, the half-angle at $A$, as determined from the above procedure, becomes negative!
In geometry, a negative quantity implies a reversal of orientation. Here that takes the form of $B$ and $C$ still lying on $l$ and $n$ as defined above, except $n$ now passes outside $\overline{AB}$ (beyond $B$) and $l$ passes outside $\overline{AC}$ (beyond $C$). Therefore, $O$ now lies outside the constructed triangle and the intended bisectors $l$ and $n$ end up bisecting exterior angles at $B$ and $C$. This reversal turns $O$ into the excenter opposite $A$ when the angle between $l$ and $n$ in the quadrantbopposite $A$ measures less than $90°$.