1. It is known that the vertex of the equilateral triangle ABC is placed at point A, and the triangle rotates around point A, and both sides of the 60-degree angle intersect with the bisector of the straight line BC and point D and ∠ACB respectively at point E. (1) When D and E are on the bisector CM of BC and ∠ACB respectively, as shown in figure/kloc-. (2) When D and E are on straight lines BC and CM respectively, as shown in Figures 2 and 3, what is the quantitative relationship among DC, Ce and AC? Please write the conclusion directly. (3) In Figure 3, when ∠ AEC = 30 and CD=4, find the length of CE.

answer

Proof: Because EAD = BAC = 60.

So ∠ bad = ∠ EAC

It is also a regular triangle ABC, so AC = AB.

Because ∠ ACB = 60 and CM is the bisector of ∠C,

So ∠ ace =1/2 (180-60) = 60.

That is ∠ ace = ∠ ACB.

So triangle ABD and triangle ACE are congruent.

So db = ce, so DC+ce = CD+BD = BC = AC.

2) Figure 2: DC-CE = AC

Figure 3: CE-CD = AC

All the proofs are to prove the congruence (ASA) of triangle ABD and triangle ACE.

3) Because ∠ ACM = 60 = ∠ B

∠BAD=∠CAE，AC=AB

So triangle ABD and triangle ACE are congruent.

So ∠ ADB = ∠ AEC = 30.

Because ∠ b = 60

So triangle ABD is a right triangle with an angle of 60,

So BD = 2ab, so BC = DC = 4.

So ce = 8

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