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If the angular bisector of an angle of a triangle bisects the opposite side, prove that the other two sides are equal.

This was the question written in my math book. I know there is a theorem related to this, but I can't understand that. I am a 9th grader so use simple language which I can understand

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Suppose $AD$ is the angle bisector of $\angle A$ in $\triangle ABC$ where $D$ is on $BC$, then $$\frac{AB}{AC} = \frac{BD}{CD}$$ so if $D$ is the midpoint of $BC$ we must have $AB=AC$. You can try to prove this important property of angle bisectors.

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  • $\begingroup$ How is AB/AC=BD/CD. If it is a theorem I have not studied it yet. Please make it still simpler. $\endgroup$ – Robin Apr 30 '17 at 11:43
  • $\begingroup$ It's called the angle bisector theorem, you can check it on wiki here: en.m.wikipedia.org/wiki/Angle_bisector_theorem $\endgroup$ – Lazy Lee Apr 30 '17 at 11:52
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This can be taken as a self evident truth (axiom). It is possible the question in your book takes a different starting point of self evident truths and wants you to use logic to derive the equality of the two lengths.

Why can this be taken as a self evident truth?

Imagine walking x steps on a straight line and then rotating 90 degrees using your left hand and then taking y steps. OK, now go back to your starting point and facing the same direction take x steps and then rotate 90 degrees using your right hand and take y steps.

Now ask yourself what is the distance you traveled in both of these journeys. Well, you should feel no more tired in either case.

The attached picture should help.

Hint: By the way you phrased the question, you are dealing with an isosceles triangle. I was forced to draw the picture the way I did. You will certainly find this helpful:

The Isosceles Decomposition Theorem

enter image description here

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  • $\begingroup$ This doesn't help. And where did you prove the two sides equal? And how did 90 degree come in the sum? $\endgroup$ – Robin May 1 '17 at 14:26

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