z = 1 + i 2 z=1+i\sqrt{2} z = 1 + i 2
p = z + 1 z = 1 + i 2 + 1 1 + i 2 = 1 + i 2 + 1 − i 2 1 2 + ( 2 ) 2 = p=z+{1\over z}=1+i\sqrt{2}+{1\over 1+i\sqrt{2}}=1+i\sqrt{2}+{1-i\sqrt{2}\over 1^2+(\sqrt{2})^2}= p = z + z 1 = 1 + i 2 + 1 + i 2 1 = 1 + i 2 + 1 2 + ( 2 ) 2 1 − i 2 =
= 3 + i 3 2 + 1 − i 2 3 = 4 + i 2 2 3 ={3+i3\sqrt{2}+1-i\sqrt{2}\over 3}={4+i2\sqrt{2}\over 3} = 3 3 + i 3 2 + 1 − i 2 = 3 4 + i 2 2
q = z − 1 z = 1 + i 2 − 1 1 + i 2 = 1 + i 2 − 1 − i 2 1 2 + ( 2 ) 2 = q=z-{1\over z}=1+i\sqrt{2}-{1\over 1+i\sqrt{2}}=1+i\sqrt{2}-{1-i\sqrt{2}\over 1^2+(\sqrt{2})^2}= q = z − z 1 = 1 + i 2 − 1 + i 2 1 = 1 + i 2 − 1 2 + ( 2 ) 2 1 − i 2 =
= 3 + i 3 2 − 1 + i 2 3 = 2 + i 4 2 3 ={3+i3\sqrt{2}-1+i\sqrt{2}\over 3}={2+i4\sqrt{2}\over 3} = 3 3 + i 3 2 − 1 + i 2 = 3 2 + i 4 2 P ( 4 3 , 2 2 3 ) , Q ( 2 3 , 4 2 3 ) , O ( 0 , 0 ) P(\dfrac{4}{3},\dfrac{2\sqrt{2}}{3}), Q(\dfrac{2}{3},\dfrac{4\sqrt{2}}{3}),O(0,0) P ( 3 4 , 3 2 2 ) , Q ( 3 2 , 3 4 2 ) , O ( 0 , 0 )
x M = x P + x Q 2 = 4 3 + 2 3 2 = 1 , x_M=\dfrac{x_P+x_Q}{2}=\dfrac{\dfrac{4}{3}+\dfrac{2}{3}}{2}=1, x M = 2 x P + x Q = 2 3 4 + 3 2 = 1 ,
y M = y P + y Q 2 = 2 2 3 + 4 2 3 2 = 2 y_M=\dfrac{y_P+y_Q}{2}=\dfrac{\dfrac{2\sqrt{2}}{3}+\dfrac{4\sqrt{2}}{3}}{2}=\sqrt{2} y M = 2 y P + y Q = 2 3 2 2 + 3 4 2 = 2
M ( 1 , 2 ) M(1, \sqrt{2}) M ( 1 , 2 )
λ = O G G M = 2 \lambda={OG\over GM}=2 λ = GM OG = 2 x G = x 0 + 2 x M 1 + 2 = 0 + 2 ⋅ 1 3 = 2 3 , x_G=\dfrac{x_0+2x_M}{1+2}=\dfrac{0+2\cdot1}{3}=\dfrac{2}{3}, x G = 1 + 2 x 0 + 2 x M = 3 0 + 2 ⋅ 1 = 3 2 ,
y G = y 0 + 2 y M 1 + 2 = 0 + 2 ⋅ 2 3 = 2 2 3 , y_G=\dfrac{y_0+2y_M}{1+2}=\dfrac{0+2\cdot\sqrt{2}}{3}=\dfrac{2\sqrt{2}}{3}, y G = 1 + 2 y 0 + 2 y M = 3 0 + 2 ⋅ 2 = 3 2 2 ,
G ( 2 3 , 2 2 3 ) G(\dfrac{2}{3},\dfrac{2\sqrt{2}}{3}) G ( 3 2 , 3 2 2 )
G P ‾ = ( x P − x G , y P − y G ) = ( 2 3 , 0 ) \overline{GP}=(x_P-x_G,y_P-y_G)=({2\over 3}, 0) GP = ( x P − x G , y P − y G ) = ( 3 2 , 0 )
G Q ‾ = ( x Q − x G , y Q − y G ) = ( 0 , 2 2 3 ) \overline{GQ}=(x_Q-x_G,y_Q-y_G)=(0, {2\sqrt{2}\over 3}) GQ = ( x Q − x G , y Q − y G ) = ( 0 , 3 2 2 )
∣ G P ‾ ∣ = 2 3 ≠ 0 , ∣ G Q ‾ ∣ = 2 2 3 ≠ 0 |\overline{GP}|={2\over 3}\not=0,|\overline{GQ}|={2\sqrt{2}\over 3}\not=0 ∣ GP ∣ = 3 2 = 0 , ∣ GQ ∣ = 3 2 2 = 0
G Q ‾ ⋅ G P ‾ = 0 = > G Q ⊥ G P \overline{GQ}\cdot\overline{GP}=0=>GQ\perp GP GQ ⋅ GP = 0 => GQ ⊥ GP Hence angle PGQ is a right angle.
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