HELP WITH PROOF. click for details and for link to diagram.?

Question

Given: Circle Z.
Seg. AE congruent Seg. DB
Conclude: Triangle ACD congruent Triangle BCE

go here for diagram. http://i11.tinypic.com/2m2e4jt.jpg

Answer 1

http://i18.tinypic.com/33adsb4.jpg
(i added stuff to you diagram)

Not sure if you need statement/reason, but here they are anyway
1.Draw Seg. ZA, ZB, ZE, ZD-2 points give a line
2.ZA=ZB=ZE=ZD--------all radii are congruent
3. ΔDZB=ΔAZE----------SSS
4. ∠1=∠2------------------CPCTC
∠DZB=∠AZE
5. ∠DZB=∠3+∠4--------Angle Addition
∠AZE=∠4+∠5
6.∠3+∠4=∠4+∠5--------Substitution
7. ∠3=∠5------------------Add. Prop
8. ΔDZA=ΔEZB----------SAS
9. AD=BE-----------------CPCTC
∠ADZ=∠BEZ
10.∠ADZ=∠1+∠6--------Angle Addition
∠BEZ=∠2+∠7
11. ∠1+∠6=∠2+∠7------Substitution
12. ∠6=∠7-----------------Add. Prop. of Equality
13.Draw DE--------------2 points give a line
14. DE=DE---------------Reflexive
15. ΔADE=ΔBED-------SSS
16. ∠AED=∠BDE--------CPCTC
17. CD=CE---------------congruent ∠'s give congruent sides
18. ΔACD=ΔBCE-------SAS

Answer 2

It's been quite a while since I dealt with proofs about geometry, but I'll give it a shot.

Since Seg AE and DB are congruent, the long leg of both triangles is congruent, as well as the short leg (hypotenuse can't be proven at this point). And angles opposite from each other made with the same 2 lines are congruent, so you can say that ACD and BCE are congruent because of the Side Angle Side formula (2 sides are congruent to their, and the angle between them is as well, making the whole triangle congruent)

Answer 3

[Edit: I just looked carefully at Ashorty's proof, and it looked good to me. In his 17th statement -- congruent angles give congruent sides -- I would've said said triangle DCE is isosceles because the base angles are congruent -- hence CD=CE. But that's a small point.

[But if Ashorty's proof is good, what was wrong with mine? I looked at that again, and found that I mis-copied the original diagram, placing the center of the circle in the wrong "quadrant" of triangles. My diagram had the center within the triangle ACD, so, in terms of the original diagram, I was trying to prove triangles ACB and DCE congruent. As I showed, that only works when AE and BD are perpendicular.

[If and when this question goes to voting, my vote goes to Ashorty. End Edit.]

Your diagram is a bit deceptive. In general, the triangles are congruent. To see this, rotate BD, holding the length constant, so that points B and E are close together (but don't let B pass beyond E). Then, BCE is a small triangle and ACD is a large triangle. They're not congruent.

Now suppose AE and BD are perpendicular. (That's what your diagram looks like.) Let F and G be the midpoints of AE and BD respectively. Draw ZF and ZG.

Then ZF and ZG are perpendicular to AE and BD respectively. Furthermore, since AE = BD, ZF = ZG. (Proof: Triangles AZE and BZD are isosceles and congruent.) Also, ZFCG is a square, so CF = CG.

Then, FE = GD. (Proof is from congruent right triangles ZFE and ZGD.) From that, CF+FE = CG+GD, so CE = CD. Finally, AE - CE = BD - CD, so AC = BC.

Therefore right triangles ACD and BCE are congruent (SAS).

But for this to hold, you have the two chords AE and BD perpendicular to each other.