(1) b = A B → , c = A C → , o = A O → \boldsymbol b=\overrightarrow{AB},\ \boldsymbol c=\overrightarrow{AC},\ \boldsymbol o=\overrightarrow{AO} b = A B , c = A C , o = A O A , B , C A,B,C A , B , C ∣ o ∣ 2 = ∣ o − b ∣ 2 , ∣ o ∣ 2 = ∣ o − c ∣ 2 . |\boldsymbol o|^2=|\boldsymbol o-\boldsymbol b|^2,\qquad |\boldsymbol o|^2=|\boldsymbol o-\boldsymbol c|^2. ∣ o ∣ 2 = ∣ o − b ∣ 2 , ∣ o ∣ 2 = ∣ o − c ∣ 2 . o ⋅ b = ∣ b ∣ 2 / 2 = 8 \boldsymbol o\cdot\boldsymbol b=|\boldsymbol b|^2/2=8 o ⋅ b = ∣ b ∣ 2 /2 = 8 o ⋅ c = ∣ c ∣ 2 / 2 = 25 / 2 \boldsymbol o\cdot\boldsymbol c=|\boldsymbol c|^2/2=25/2 o ⋅ c = ∣ c ∣ 2 /2 = 25/2 o = u b + v c \boldsymbol o=u\boldsymbol b+v\boldsymbol c o = u b + v c 16 u + 5 v = 8 , 5 u + 25 v = 25 2 . \displaystyle
16u+5v=8,\qquad 5u+25v=\frac{25}{2}. 16 u + 5 v = 8 , 5 u + 25 v = 2 25 . u = 11 / 30 , v = 32 / 75 u=11/30,\ v=32/75 u = 11/30 , v = 32/75 A O → ⋅ A B → = 8 \overrightarrow{AO}\cdot\overrightarrow{AB}=8 A O ⋅ A B = 8 A O → = 11 30 A B → + 32 75 A C → . \displaystyle
\overrightarrow{AO}=\frac{11}{30}\overrightarrow{AB}+\frac{32}{75}\overrightarrow{AC}. A O = 30 11 A B + 75 32 A C .
(2) 底辺 B C BC B C D B C DBC D B C D D D B C BC B C B C BC B C B D BD B D D C DC D C ∠ B A D = ∠ D A C \angle BAD=\angle DAC ∠ B A D = ∠ D A C A D AD A D ∠ A \angle A ∠ A B E : E C = A B : A C = 4 : 5 BE:EC=AB:AC=4:5 B E : E C = A B : A C = 4 : 5 A E → = 5 9 A B → + 4 9 A C → , B E = 4 31 9 , E C = 5 31 9 , \displaystyle
\overrightarrow{AE}=\frac59\overrightarrow{AB}+\frac49\overrightarrow{AC},\quad BE=\frac{4\sqrt{31}}9,\quad EC=\frac{5\sqrt{31}}9, A E = 9 5 A B + 9 4 A C , B E = 9 4 31 , E C = 9 5 31 , B C 2 = 4 2 + 5 2 − 2 ⋅ 5 = 31 BC^2=4^2+5^2-2\cdot5=31 B C 2 = 4 2 + 5 2 − 2 ⋅ 5 = 31 cos B = A B 2 + B C 2 − A C 2 2 A B ⋅ B C = 11 4 31 . \displaystyle
\cos B=\frac{AB^2+BC^2-AC^2}{2AB\cdot BC}=\frac{11}{4\sqrt{31}}. cos B = 2 A B ⋅ B C A B 2 + B C 2 − A C 2 = 4 31 11 . A B E ABE A B E A E 2 = 4 2 + ( 4 31 9 ) 2 − 2 ⋅ 4 ⋅ 4 31 9 ⋅ 11 4 31 = 1000 81 . \displaystyle
AE^2=4^2+\left(\frac{4\sqrt{31}}9\right)^2-2\cdot4\cdot\frac{4\sqrt{31}}9\cdot\frac{11}{4\sqrt{31}}=\frac{1000}{81}. A E 2 = 4 2 + ( 9 4 31 ) 2 − 2 ⋅ 4 ⋅ 9 4 31 ⋅ 4 31 11 = 81 1000 . K K K A D , B C AD,BC A D , B C E E E E A ⋅ E D = E B ⋅ E C EA\cdot ED=EB\cdot EC E A ⋅ E D = E B ⋅ E C A − E − D A-E-D A − E − D t = A D A E = 1 + E D E A = 1 + B E ⋅ C E A E 2 = 1 + 620 1000 = 81 50 . \displaystyle
t=\frac{AD}{AE}=1+\frac{ED}{EA}=1+\frac{BE\cdot CE}{AE^2}=1+\frac{620}{1000}=\frac{81}{50}. t = A E A D = 1 + E A E D = 1 + A E 2 B E ⋅ C E = 1 + 1000 620 = 50 81 .
(3) b = A B → , c = A C → \boldsymbol b=\overrightarrow{AB},\boldsymbol c=\overrightarrow{AC} b = A B , c = A C λ = B P / B C \lambda=BP/BC λ = B P / B C A P → = ( 1 − λ ) b + λ c \overrightarrow{AP}=(1-\lambda)\boldsymbol b+\lambda\boldsymbol c A P = ( 1 − λ ) b + λ c A O → = 11 30 b + 32 75 c , A D → = 81 50 ( 5 9 b + 4 9 c ) = 9 10 b + 18 25 c . \displaystyle
\overrightarrow{AO}=\frac{11}{30}\boldsymbol b+\frac{32}{75}\boldsymbol c,\qquad
\overrightarrow{AD}=\frac{81}{50}\left(\frac59\boldsymbol b+\frac49\boldsymbol c\right)=\frac9{10}\boldsymbol b+\frac{18}{25}\boldsymbol c. A O = 30 11 b + 75 32 c , A D = 50 81 ( 9 5 b + 9 4 c ) = 10 9 b + 25 18 c . b ⋅ b = 16 , c ⋅ c = 25 , b ⋅ c = 5 \boldsymbol b\cdot\boldsymbol b=16,\ \boldsymbol c\cdot\boldsymbol c=25,\ \boldsymbol b\cdot\boldsymbol c=5 b ⋅ b = 16 , c ⋅ c = 25 , b ⋅ c = 5 P O → ⋅ P D → = 0 \overrightarrow{PO}\cdot\overrightarrow{PD}=0 P O ⋅ P D = 0 31 λ 2 − 31 λ + 31 5 = 0 , λ = 5 ± 5 10 . \displaystyle
31\lambda^2-31\lambda+\frac{31}{5}=0,
\quad\lambda=\frac{5\pm\sqrt5}{10}. 31 λ 2 − 31 λ + 5 31 = 0 , λ = 10 5 ± 5 . B P > C P BP>CP B P > C P λ > 1 / 2 \lambda>1/2 λ > 1/2 λ = ( 5 + 5 ) / 10 \lambda=(5+\sqrt5)/10 λ = ( 5 + 5 ) /10 C P B P = 1 − λ λ = 3 − 5 2 . \displaystyle
\frac{CP}{BP}=\frac{1-\lambda}{\lambda}=\frac{3-\sqrt5}{2}. B P C P = λ 1 − λ = 2 3 − 5 .
