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Let triangle $\Delta ABC$ have $AB=AC$. Then we draw the angle bisector from $B$ to $AC$ intersecting at $D$. Find the angle $\angle BAC$ if $BC=AD+BD$.

My attempts:

I know that the answer is 100° but I couldn't prove that if you extended $AD$ to a point $E$ so it is equal to $BC$, then the angle $\angle DCE$ is the same as $\angle ACB$.

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    $\begingroup$ I solved your problem. If you want to see my solution show us your trying. $\endgroup$ – Michael Rozenberg Mar 28 at 10:22
  • $\begingroup$ I know that the answer is 100° but I couldn't prove that if you extended AD to a point E so it is equal to BC the angle DCE is the same as ACB. $\endgroup$ – AJMC2002 Mar 28 at 17:18
  • $\begingroup$ I posted. See now. $\endgroup$ – Michael Rozenberg Mar 28 at 17:26
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In the standard notation we obtain: $$\frac{AD}{DC}=\frac{AB}{BC}=\frac{c}{a}$$ and $$AD+DC=AC=b,$$ which gives $$AD=\frac{bc}{a+c},$$ $$DC=\frac{ab}{a+c}$$ and $$BD^2=AB\cdot BC-AD\cdot DC=ac-\frac{b^2ac}{(a+c)^2}=ab-\frac{ab^3}{(a+b)^2},$$ which gives $$BD=\frac{a\sqrt{b(a+2b)}}{a+b}.$$ Id est, by the given we obtain: $$a=\frac{b^2}{a+b}+\frac{a\sqrt{b(a+2b)}}{a+b}$$ or $$a\sqrt{b(a+2b)}=a^2+ab-b^2.$$ Now, $\sin\frac{\alpha}{2}=\frac{\frac{a}{2}}{b}=\frac{a}{2b}.$

Let $\sin\frac{\alpha}{2}=x.$

Thus, $a=2xb$ and we obtain: $$2x\sqrt{2(x+1)}=4x^2+2x-1$$ or $$8x^2(x+1)=(4x^2+2x-1)^2,$$ where $x>0$ and $4x^2+2x-1>0.$

We obtain: $$16x^4+8x^3-12x^2-4x+1=0$$ or $$8x^3(2x+1)-12x^2-6x+2x+1=0$$ or $$(2x+1)(8x^3-6x+1)=0$$ or $$3x-4x^3=\frac{1}{2}$$ or $$\sin\frac{3\alpha}{2}=\frac{1}{2},$$ which gives $$\frac{3\alpha}{2}=30^{\circ},$$ which is impossible because $4x^2+2x-1>0,$ or $$\frac{3\alpha}{2}=150^{\circ},$$ which gives $\alpha=100^{\circ}$ and $\beta=\gamma=40^{\circ}.$

The fact that $$BD^2=AB\cdot BD-AD\cdot DC$$ we can prove by the following reasoning.

Let $\Phi$ be a circumcircle of $\Delta ABC$ and $AD\cap\Phi=\{A,E\}$.

Thus, $\measuredangle BAC=\measuredangle BEC$ and $\measuredangle ABD=\measuredangle EBD,$ which gives $\Delta ABD\sim\Delta EBC.$

Hence, $$\frac{AB}{BE}=\frac{BD}{BC}$$ or $$AB\cdot BC=BD\cdot BE$$ or $$AB\cdot BC=BD(BD+DE)$$ or $$BD^2=AB\cdot BC-BD\cdot DE$$ or $$$BD^2=AB\cdot BC-AD\cdot DC.$$

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  • $\begingroup$ Could you elaborate on how did you get BD²=AB•BC-AD•DC? Thanks $\endgroup$ – AJMC2002 Mar 29 at 5:39
  • $\begingroup$ @user658009 I added something. See now. $\endgroup$ – Michael Rozenberg Mar 29 at 8:13

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