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
2./view/8 afab 0 c 38 BD 63 186 BCE BBC 43 . html
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