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In $\triangle ABC$, $\angle B$ is a right angle. $D$ and $E$ are points on segment $AC$ such that $AD:DE:EC = 1:2:\sqrt{3}$. Then, prove that $\angle DBE = 45^\circ$.enter image description here

Harish Chandra Rajpoot
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Yashbhatt
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1 Answers1

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EDIT2: OP is right. there are two positions which have $45 $ degree.I draw 4 pictures to show it.

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enter image description here

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one position is show on 2nd picture,which can be proved by following solution.

EDIT: here is the solution:

let $AB=BC=1,AC=\sqrt{2},AD=\dfrac{AC}{1+2+\sqrt{3}}=\dfrac{\sqrt{2}}{3+\sqrt{3}}=\dfrac{3\sqrt{2}-\sqrt{6}}{6}=\dfrac{\sqrt{2}}{2}-\dfrac{\sqrt{6}}{6}$

$M$ is midpoint of $AC,AM=BM=\dfrac{\sqrt{2}}{2},DM=BM-AD=\dfrac{\sqrt{2}}{2}-\dfrac{\sqrt{2}}{2}+\dfrac{\sqrt{6}}{6}=\dfrac{\sqrt{6}}{6}$

$\dfrac{BM}{DM}=\sqrt{3} \implies \angle BDM= \dfrac{\pi}{3}$

let $N$ is on $BC$, and $EN \perp BC$

now we need to prove $\dfrac{EN}{BN}=\dfrac{\sqrt{3}}{3}$

this is left for op.

EDIT 2: another position is shown on 4th picture, green triangle. unfortunately it is not a "normal" triangle and the length has $\sqrt{\sqrt{p}+q}$ style.

chenbai
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  • And if the given triangle is isoscles, then we just have to prove congruence, right? – Yashbhatt Apr 06 '14 at 09:16
  • I don't know where congruence is. My approach will be find $\angle BDE ,\angle EBC$ – chenbai Apr 06 '14 at 09:37
  • I wonder how we can use that to show the angle as 45. – Yashbhatt Apr 06 '14 at 09:50
  • @Yashbhatt, I put part of solution and think you can finish it. – chenbai Apr 06 '14 at 12:37
  • You probably didn't realize that this question is the same as the one you previously answered. Have a look at this. – Yashbhatt Apr 06 '14 at 15:30
  • Ya,I just found that problem is more general.I do forget it. :) . – chenbai Apr 07 '14 at 06:12
  • That solution seems more appealing than this one. – Yashbhatt Apr 07 '14 at 06:57
  • @chenbai Nice work. Up-voted. However, I wonder can the result be arrived directly. That is, proving that the angle DBE will never be 45 degrees unless AB = BC. [A make-up example is …. After some work …. sin DBE = √2AB / (2BC), then angle DBE = 45 only when AB = BC.] P.S. This comment can be ignored. – Mick Apr 08 '14 at 03:54
  • @Mick,thanks for your comments and it reminder me there is a mistake and I just correct it. there are two solution for this question. – chenbai Apr 09 '14 at 03:12