Geometric Proof?

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f1f2f3

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I have this geometry problem here which I am having difficulties with (Attached)
I am finding it hard to prove that BD=CD.
I am trying to use congruent triangles for proof and I only need to prove that line AD is perpendicular to line BC to be able to prove that BD=CD, however I'm not quite sure how to do this.
 

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Prove that triangle BEA is congruent to triangle CEA by SSS.
That will give you sufficient set of congruent angles to prove that triangle BED is congruent to triangle CED by ASA.
 
Or, once you have proved that triangle BEA is congruent to triangle CEA as suggested, you can readily prove that triangle BAD is congruent to triangle CAD by SAS....
 
But then how do I prove that line AE is straight and divides triangles BEA and CEA into 2 congruent triangles?
 
f1f2f3 said:
But then how do I prove that line AE is straight

What do you mean? Line AE joins two terminal points A and E - it does not have a choice of not being straight.


and divides triangles BEA and CEA into 2 congruent triangles?
Click to expand...
 
Sorry, I meant, how do you know the line makes the two triangles congruent? It could be on a slope so that one triangle is larger than the other. How do I know that it goes straight down to divide the triangles into two congruent ones?
Thanks
 
f1f2f3 said:
Sorry, I meant, how do you know the line makes the two triangles congruent? It could be on a slope so that one triangle is larger than the other. How do I know that it goes straight down to divide the triangles into two congruent ones?
Thanks
Click to expand...

You are GIVEN that AB = AC, and that EB = EC. Now, that is two pairs of equal sides in triangles ABE and ACE, right? The third side in each triangle is AE....and AE = AE by the reflexive property of equality. You have three sides of one triangle equal in length to the three sides of the second triangle. The Side-Side-Side congruence postulate (or theorem, depending on your textbook) says that "If three sides of one triangle are equal (or congruent) to the three sides of another triangle, the two triangles are congruent."

Because segment AE is a side of each of the triangles, it should (obviously, I hope) be the same length in each of them.
 
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