A A 1 {\displaystyle AA_{1}} B B 1 {\displaystyle BB_{1}} C C 1 {\displaystyle CC_{1}} △ A B B 1 {\displaystyle \triangle ABB_{1}} A C 1 → C 1 B → ⋅ B O → O B 1 → ⋅ B 1 C → O A → = − 1 {\displaystyle {\frac {\overrightarrow {AC_{1}}}{\overrightarrow {C_{1}B}}}\cdot {\frac {\overrightarrow {BO}}{\overrightarrow {OB_{1}}}}\cdot {\frac {\overrightarrow {B_{1}C}}{\overrightarrow {OA}}}=-1} △ B C B 1 {\displaystyle \triangle BCB_{1}}
B 1 O → O B → ⋅ B A 1 → A 1 C → ⋅ C A → A B 1 → = − 1 {\displaystyle {\frac {\overrightarrow {B_{1}O}}{\overrightarrow {OB}}}\cdot {\frac {\overrightarrow {BA_{1}}}{\overrightarrow {A_{1}C}}}\cdot {\frac {\overrightarrow {CA}}{\overrightarrow {AB_{1}}}}=-1}
( A C 1 → C 1 B → ⋅ B O → O B 1 → ⋅ B 1 O → O A → ) ⋅ ( B 1 O → O B → ⋅ B A 1 → A 1 O → ⋅ C A → A B 1 → ) = ( − 1 ) ⋅ ( − 1 ) {\displaystyle \left({\frac {\overrightarrow {AC_{1}}}{\overrightarrow {C_{1}B}}}\cdot {\frac {\overrightarrow {BO}}{\overrightarrow {OB_{1}}}}\cdot {\frac {\overrightarrow {B_{1}O}}{\overrightarrow {OA}}}\right)\cdot \left({\frac {\overrightarrow {B_{1}O}}{\overrightarrow {OB}}}\cdot {\frac {\overrightarrow {BA_{1}}}{\overrightarrow {A_{1}O}}}\cdot {\frac {\overrightarrow {CA}}{\overrightarrow {AB_{1}}}}\right)=(-1)\cdot (-1)}
A C 1 → C 1 B → ⋅ B A 1 → A 1 C → ⋅ C B 1 → B 1 A → = 1 {\displaystyle {\frac {\overrightarrow {AC_{1}}}{\overrightarrow {C_{1}B}}}\cdot {\frac {\overrightarrow {BA_{1}}}{\overrightarrow {A_{1}C}}}\cdot {\frac {\overrightarrow {CB_{1}}}{\overrightarrow {B_{1}A}}}=1}
A C 1 → C 1 B → ⋅ B A 1 → A 1 C → ⋅ C B 1 → B 1 A → = − 1 {\displaystyle {\frac {\overrightarrow {AC_{1}}}{\overrightarrow {C_{1}B}}}\cdot {\frac {\overrightarrow {BA_{1}}}{\overrightarrow {A_{1}C}}}\cdot {\frac {\overrightarrow {CB_{1}}}{\overrightarrow {B_{1}A}}}=-1}
A C 1 → ↑↑ C 1 B → → A C 1 → C 1 B → > 0 B A 1 → ↑↑ A 1 C → → B A 1 → A 1 C → > 0 C B 1 → ↑↓ B 1 A → → C B 1 → B 1 A → > 0 } A C 1 → C 1 B → ⋅ B A 1 ′ → A 1 ′ C → ⋅ C B 1 → B 1 A → < 0 {\displaystyle \left.{\begin{matrix}{\overrightarrow {AC_{1}}}\uparrow \uparrow {\overrightarrow {C_{1}B}}\to {\frac {\overrightarrow {AC_{1}}}{\overrightarrow {C_{1}B}}}>0\\{\overrightarrow {BA_{1}}}\uparrow \uparrow {\overrightarrow {A_{1}C}}\to {\frac {\overrightarrow {BA_{1}}}{\overrightarrow {A_{1}C}}}>0\\{\overrightarrow {CB_{1}}}\uparrow \downarrow {\overrightarrow {B_{1}A}}\to {\frac {\overrightarrow {CB_{1}}}{\overrightarrow {B_{1}A}}}>0\end{matrix}}\right\}{\frac {\overrightarrow {AC_{1}}}{\overrightarrow {C_{1}B}}}\cdot {\frac {\overrightarrow {BA_{1}'}}{\overrightarrow {A'_{1}C}}}\cdot {\frac {\overrightarrow {CB_{1}}}{\overrightarrow {B_{1}A}}}<0} △ A K C 1 ∼ △ B M C 1 {\displaystyle \triangle AKC_{1}\sim \triangle BMC_{1}} A C 1 B C 1 = A K B M {\displaystyle {\frac {AC_{1}}{BC_{1}}}={\frac {AK}{BM}}}
△ B M A 1 ∼ △ C N A 1 {\displaystyle \triangle BMA_{1}\sim \triangle CNA_{1}} B A 1 C A 1 = B M C N {\displaystyle {\frac {BA_{1}}{CA_{1}}}={\frac {BM}{CN}}}
△ B 1 C N ∼ △ B 1 A K {\displaystyle \triangle B_{1}CN\sim \triangle B_{1}AK} B 1 C B 1 A = C N A K {\displaystyle {\frac {B_{1}C}{B1_{A}}}={\frac {CN}{AK}}}
( 1 ) {\displaystyle (1)} ( 1 ) , ( 2 ) → B A 1 → A 1 C → = B A 1 → A 1 C → {\displaystyle (1),(2)\to {\frac {\overrightarrow {BA_{1}}}{\overrightarrow {A_{1}C}}}={\frac {\overrightarrow {BA_{1}}}{\overrightarrow {A_{1}C}}}}
A K → K D → ⋅ D O → O B → ⋅ B M → M A → = 1 ⋅ A B B C ⋅ ( B C A D ) = ( − 1 ) {\displaystyle {\frac {\overrightarrow {AK}}{\overrightarrow {KD}}}\cdot {\frac {\overrightarrow {DO}}{\overrightarrow {OB}}}\cdot {\frac {\overrightarrow {BM}}{\overrightarrow {MA}}}=1\cdot {\frac {AB}{BC}}\cdot \left({\frac {BC}{AD}}\right)=(-1)}
D O → O B → ⋅ B N → N C → ⋅ C M → M D → = A B B C ⋅ 1 ⋅ ( − B C A D ) = ( − 1 ) {\displaystyle {\frac {\overrightarrow {DO}}{\overrightarrow {OB}}}\cdot {\frac {\overrightarrow {BN}}{\overrightarrow {NC}}}\cdot {\frac {\overrightarrow {CM}}{\overrightarrow {MD}}}={\frac {AB}{BC}}\cdot 1\cdot \left(-{\frac {BC}{AD}}\right)=(-1)}
B B 2 ( ( ⋅ ) B 2 ϵ [ A C ] ) {\displaystyle BB_{2}((\cdot )B_{2}\epsilon [AC])} ( ⋅ ) O {\displaystyle (\cdot )O}
C B 1 → B 1 A → = C B 2 → B 2 A → {\displaystyle {\frac {\overrightarrow {CB_{1}}}{\overrightarrow {B_{1}A}}}={\frac {\overrightarrow {CB_{2}}}{\overrightarrow {B_{2}A}}}}
A C 1 → C 1 B → ⋅ B A 1 → A 1 C → ⋅ C B 1 → B 1 A → = 1 ⋅ 1 ⋅ 1 = 1 {\displaystyle {\frac {\overrightarrow {AC_{1}}}{\overrightarrow {C_{1}B}}}\cdot {\frac {\overrightarrow {BA_{1}}}{\overrightarrow {A_{1}C}}}\cdot {\frac {\overrightarrow {CB_{1}}}{\overrightarrow {B_{1}A}}}=1\cdot 1\cdot 1=1}
A C 1 → C 1 B → ⋅ B A 1 → A 1 C → ⋅ C B 1 → B 1 A → = A C 1 C 1 B ⋅ B A 1 A 1 C ⋅ C B 1 B 1 A = A C A B ⋅ A B A C ⋅ B C A B = 1 {\displaystyle {\frac {\overrightarrow {AC_{1}}}{\overrightarrow {C_{1}B}}}\cdot {\frac {\overrightarrow {BA_{1}}}{\overrightarrow {A_{1}C}}}\cdot {\frac {\overrightarrow {CB_{1}}}{\overrightarrow {B_{1}A}}}={\frac {AC_{1}}{C_{1}B}}\cdot {\frac {BA_{1}}{A_{1}C}}\cdot {\frac {CB_{1}}{B_{1}A}}={\frac {AC}{AB}}\cdot {\frac {AB}{AC}}\cdot {\frac {BC}{AB}}=1}
A C 1 → C 1 B → ⋅ B A 1 → A 1 C → ⋅ C B 2 → B 2 A → = 1 {\displaystyle {\frac {\overrightarrow {AC_{1}}}{\overrightarrow {C_{1}B}}}\cdot {\frac {\overrightarrow {BA_{1}}}{\overrightarrow {A_{1}C}}}\cdot {\frac {\overrightarrow {CB_{2}}}{\overrightarrow {B_{2}A}}}=1}
4 S △ A B C {\displaystyle 4S\triangle ABC} B D A D = B C A C {\displaystyle {\frac {BD}{AD}}={\frac {BC}{AC}}} △ A B D {\displaystyle \triangle ABD} △ A D C {\displaystyle \triangle ADC} ( 1 ) ( 2 ) : B D s i n α = A B s i n β C D s i n α = A C s i n β {\displaystyle {\frac {(1)}{(2)}}:{\frac {{\frac {BD}{sin\alpha }}={\frac {AB}{sin\beta }}}{{\frac {CD}{sin\alpha }}={\frac {AC}{sin\beta }}}}} B D C D = A B A C {\displaystyle {\frac {BD}{CD}}={\frac {AB}{AC}}} ∠ A M D = ∪ A D + ∪ B C 2 {\displaystyle \angle AMD={\frac {\cup AD+\cup BC}{2}}} ( ⋅ ) O ; R {\displaystyle (\cdot )O;R} A C ∩ B D = ( ⋅ ) M {\displaystyle AC\cap BD=(\cdot )M} ∠ B D C = 1 2 ∪ B C {\displaystyle \angle BDC={\frac {1}{2}}\cup BC} ∠ A C D = 1 2 ∪ A D {\displaystyle \angle ACD={\frac {1}{2}}\cup AD} △ M D C {\displaystyle \triangle MDC} ∠ A M D {\displaystyle \angle AMD} A C = d , ∠ C A D = α {\displaystyle AC=d,\angle CAD=\alpha } S a b c d {\displaystyle S_{abcd}} △ A C D ( ∠ D = 90 ∘ ) {\displaystyle \triangle ACD(\angle D=90^{\circ })} C D A C = s i n α → C D = d ⋅ s i n α {\displaystyle {\frac {CD}{AC}}=sin\alpha \to CD=d\cdot sin\alpha } A D A C = c o s α → A D = d ⋅ c o s α {\displaystyle {\frac {AD}{AC}}=cos\alpha \to AD=d\cdot cos\alpha } S a b c d = A D ⋅ C D = d 2 ⋅ s i n α ⋅ c o s α = d 2 ⋅ s i n 2 α 2 {\displaystyle S_{abcd}=AD\cdot CD=d^{2}\cdot sin\alpha \cdot cos\alpha ={\frac {d^{2}\cdot sin2\alpha }{2}}} S a b c d = d 2 ⋅ s i n 2 α 2 {\displaystyle S_{abcd}={\frac {d^{2}\cdot sin2\alpha }{2}}}
![{\displaystyle BB_{2}((\cdot )B_{2}\epsilon [AC])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/194fd57ef0a7c8cc50ab86b604532a2a6eedb526)
