(1) expresses the value of ∠DME with the algebraic expression of α;
(2) If point M moves on ray BC (not coincident with point D), and other conditions remain unchanged, does the size of ∠DME change with the position of point M? Please draw a picture, give your conclusion and explain the reasons.
Solution: (1) Solution 1: Make a straight line EM intersect AB at point F, and the extension line intersect AC at point G (see figure 1).
∫AD splitting∠ ∠BAC,
∴∠ 1 = ∠ 2.( 1)
∵ME⊥AD,
∴∠AEF=∠AEG=90
∴∠3=∠G.
∠∠3 =∠b+∠DME,
∴∠ACB=∠G+∠GMC=∠G+∠DME,
∴∠B+∠DME=∠ACB-∠DME.
∴∠dme= 1 2(≈ACB-∠b)=α2; (2 points)
Scheme 2: As shown in Figure 2 (without auxiliary lines),
∫AD splitting∠ ∠BAC,
∴∠ 1 = ∠ 2.( 1)
∵ME⊥AD,
∴∠DEM=90,∠ADC+∠DME=90。
∠∠ADB =∠2+∠C = 90+∠DME,
∴∠DME=∠2+∠C-90。
∫∠ADC =∠ 1+∠B,
∴∠ 1=∠ADC-∠B.
∴∠dme=∠ 1+∠c-90 =(∠ADC-∠b)+∠c-90
=∠C-∠B-(90 -∠ADC)=∠C-∠B-∠DME
∴∠dme= 1 2(∠c-∠b)=α2; (2 points)
(2) As shown in Figures 3 and 4, the size of ∠DME remains unchanged when point M moves on the ray BC (not coincident with point D) (the same is true when point M moves to points B and C).
Prove 1: set point m moves to m', and make m' e' ⊥ ad at point e' through point m'
∵m′e′⊥ad,
∴me∥m′e′.
∴∠ DM' e' = ∠ DME = α 2。 (4 points)
By the way, he said that the map I copied was illegal, and there was nothing I could do.