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Point $D$ is chosen on side $BC$ of $\Delta ABC$ such that the incircles of $\Delta ACD$ and $\Delta ABD $ are tangent at $G$. Let line $l$ be the angle bisector of $\angle ABC$ ,line $m$ be the angle bisector of $\angle ACB$,and line $n$ be the perpendicular to $BC$ at point $D$.

Prove that lines $l,m$ and $n$ are concurrent.

My attempt:

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I let $ CR$ (angle bisector of $C$) and $DX$ ( line perpendicular to $BC$ at $D$) meet at $I$ ,then ,considering $\Delta CRA$ ,i have to prove that points $B,I,Q$ (where $BQ$ is the angle bisector of $B$) are collinear. For that i've considered $\Delta CRA$ from which i have by Menelaus's Theorem (given that $I,Q,B$ are points that lie on the sides of this triangle)that: $$\cfrac{AB \cdot CQ \cdot RI}{RB \cdot AQ \cdot CI}=1 $$

Applying the Angle Bisector Theorem to $BQ$ i have that $$\frac {CQ}{AQ}=\cfrac{BC}{AB} \tag 1$$ ,doing the same for angle bisector $CR$ i have : $$ BR \cdot AC = AR \cdot BC $$ Multiplying this relation with $(1)$, i have that $\cfrac {AB}{RB}=\cfrac {AB \cdot AC}{BC \cdot AR}$ ,so to prove that the lines are concurrent i have to prove that $\cfrac {RI}{CI}=\cfrac {AR}{AC}$

I am having some trouble proving this,can you give me some hints ?

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    $\begingroup$ The thing you must prove is very obvious just think that I is the intersection of the bisector in A to RC so RI RC AC and AR also have the bisector theorem so it is true $\endgroup$ – Weijie Chen Nov 4 '15 at 23:49
  • $\begingroup$ Indeed you're correct,i didn't see that.Thank you $\endgroup$ – Nameless Nov 5 '15 at 10:11
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Let the tangent of circle $O$ at $AB$ be $M$, also let $O$'s tangent at $BD$ be $X$.

Let the tangent of circle $O_1$ at $AC$ be $N$, also let $O_1$'s tangent at $DC$ be $Y$.

Then obviously $IX=IM$ and $IY=IN$ as triangles $IXB$ and $IMB$ are congruent and $IY$ part similarly.

Connect $AI$, since $I$ is the incenter of $ABC$, $AI$ is also an angle bisector. Using the similar congruence over triangles $AIM$ and $AIN$ we have $IM=IN$.

Hence $IX=IY$. But $XD=DY$ as they are both equal to $DG$.

This is enough to show $ID$ is a median of isoceles triangle $IXY$ and hence $ID\perp XY$ or in other words $ID\perp BC$.

Hence $ID$ is the same line as your line $n$. QED.

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  • $\begingroup$ Let me take some time to fully read this,then i 'll come back . $\endgroup$ – Nameless Nov 4 '15 at 10:08
  • $\begingroup$ Back.Indeed that's a nice way to solve the problem.Just out of curiosity,how did you come up with all of that in a matter of few minutes since i had posted the question ? $\endgroup$ – Nameless Nov 4 '15 at 10:29
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    $\begingroup$ In terms of coming up with the solution, you have to go backward by first realizing $DX=DY$ and conclude the problem is equivalent to proving $IXY$ is isosceles. $\endgroup$ – cr001 Nov 4 '15 at 10:42
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In my solution I will call the tangents of the circles, $Q$ the tangent of the incircle of $\Delta ABD$ to $AB$ and $M$ to $BC$, N the tangent of incircle of $\Delta ADC$ to $BC$ and $S$ to $AC$.

Fistly notice that $AG$ is the radical axis from the two circles, so it is obvious that $AQ^2=AG^2=AS^2$, hence $AQ=AG=AS$.

Let now call $z=BM=BQ$ because they are tangent to the incircle $ABD$, for the same reasoning $y=CN=CS$ and $t=AQ=AS$, $x=DM=DN$ because $D$ is on the common tangent of the two circles and it is on the radical axis.

Therefore $z+2x+y=a$, $z+t=c$, $t+y=b$.

By summing all the equations we get $2z+2x+2y+2t=a+b+c$, and $z+x+y+t=\cfrac{a+b+c}{2}$. Now observe that $y+t=b$, so by substiting it we get that $z+x=\cfrac{a+c-b}{2}$. So the point D is the point of tangency from the incircle of $\Delta ABC$ and therefore it the incenter of $ABC$ is perpendicular to it.

Q.E.D

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  • $\begingroup$ Thanks for your answer, it's always good to have different points of view on a problem. $\endgroup$ – Nameless Nov 5 '15 at 10:16
  • $\begingroup$ Now that i've understood your solution,i can say that it's really beautiful as well.Unfortunately i am not allowed to choose two accepted answers ...One question: have you also worked backwards ,i.e you already knew that would be true and then you proved it ? $\endgroup$ – Nameless Nov 5 '15 at 10:42
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    $\begingroup$ My idea was that D was the tangent point from the incicle to the side BC if it is perpendicular so yes I could be backward reasoning $\endgroup$ – Weijie Chen Nov 5 '15 at 11:38
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    $\begingroup$ I mean, the intuition told me that there is only one line so then I tried to prove that. Because it is easy to see that D is the tangency point $\endgroup$ – Weijie Chen Nov 5 '15 at 11:48

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