[r]
(1)∠ ! ∠ " !
#$ % &' ( ∆
⇒ ) ⇒∠ ∠
* ⇒ ∠ *∠ +,- / &0 12 567∠ ∠ * ∠
∠ 89!!: + !;" !4 89!!: 88!! !! ∠ *∠ *∠
⇒ * * +,- / &0 12
(2)# ( 12 => @A71 B C =C =< ⇒∆ ) 1D C
⇒∠=C< ∠ =<C +84
# <= / <> &0 <C C ∠><C *∠=<C +E4
#$ +84 &0 +E4 ∠><C *∠=C< FG C>< ∠><C *∠>C<
-7 C> *>< +,- / &0 12 +"4 C> => ;=C => ;=< +H4
#$ +"4 &0 +H4 => ;=< *><
< =><I∠= J!! # ( 12 =< @A7C =C K=< L2 > &M C < ? >< *>C
# ( =< @A71 B N =N =C # =N K=< +5%=C K=<4
L( N / = &0 < O0 >= ⊥ CN &0 C= =N
(3)# ∠>N= *∠><N +∠>N= @0 0 >N<4 C ∠>C< *∠><C
FG >C< ∠>C< *∠><C
-7 >< *>C +,- / &0 12 P
< =>< => K=<IO @0 - B >< ∠>=O &0 ∠O=<
5' 12 O= &0 ( @A71 B C OC O= FG O=> &0 OC<
O= OC
∠=O> ∠CO< +12 1D 4 O> O< +O @0 #Q ><4 C ∆ =∆ +
-7 => <C ∠>=O ∠OC< # => <C => K=< ⇒ <C K<= FG =C< <C K=<
⇒ ∠O=< K∠OC< +,- / &0 12 O0 ∠O=< K∠OC< &0∠>=O ∠OC<
-7 ∠O=< K∠>=O
(4)VC ⊥ >< ∆ =
∆ + -7W : X ⇒ =C C
∆ &-R ⇒ C KC<
∆ + &-R Y Z -7W -7 =C KC<
"
< % &' ( => *=< < ? N> *N<
< % &G (
< ? >C ;<N K=> ;=<
=> *=< ⇒ > * <+1[\ ]( @M Z %1[\ -@M Z
> * < ⇒ N> *N<
(5)# =CN &-R N -7 <N K=< -7 >C ;<N K=> ;=<
< =><IO @0 - B >< < ? => ;=< *E=O
# ( 12 O= @A71 B C OC O= FG ∆O=> &0 ∆OC<
O= OC
∠=O> ∠CO< +12 1D O> O< +
C ∆ =∆ + ⇒ => C<
FG =C< <C ;=< *=C + A 1^ ? C => ;=< *=C =C E=O
-7 => ;=< *E=O
# < =>< T ) C U =< C 1P30 => &0 ><I >C< _
(6)# a b ∠>C< _ ∠C8*J!! ⇔ E∠C8*89!!
# =>C ∠C8 ∠= ;∠>E +84
# ><C ∠C8;∠>8;∠<8 89!! +E4 <R a & +84 &0 +E4 1[c
E∠C8;∠>8;∠< ∠= ;∠>E;89!! ⇔ ∠= :∠< E∠C8: 89!!*! +∠C8 _4
⇒ ∠= *∠< ⇔ >< *=>
$ < =><IO @0 B < ? O> ;O< K=> ;=<
5'1[\ ^ >O U =< C
5%O S =>< ( C / = &0 < -7 =< =C ;C<
FG =>C C> K=> ;=C + A 1^ ? ⇒O> ;OC K=> ;=C +84
FG OC< O< KC< ;OC +E4 + A 1^ ? <R +84 &M +E4 & &M &
O> ;O< ;OC K=> ;=C ;C< ;OC
⇒ O> ;O< K=> ;+=C ;C<4 ⇒O> ;O< K=> ;=<
% < =>< => *=< =C @0 T ) >=< +C ∈ ><4 O @0 B ( ^ =C
(7)# ( => @A71 B N =N =< &%=> *=<I ( N / = &0 >
-7 =N ;N> =>
⇒ N> => : =N => d =< FG ∆=NO &0 ∆=<O =N =<
∠N=O ∠<=O +=C @0 T ) >=<4 =O
-C ∆ =∆ + -7 ON O<
FG ON> O> : ON KN> + A 1^ ? C O> : O< K=> : =<
< =>< = J!! # ( =>I=< @` @[c @A7 1 B C &0 N < ? CN K><
L2 C &0 < =NI=< @` @[c @0 % - % ]( CNIC< ( 1[\ ^ =<
0 =N K=N +5%N -P =<4
-7 CN KC< +,- / 1[\ ]( &0 % -
(8)-1[\ ]( C<I>< ( 1[\ ^ => =C K=> +C -P =>4 -7 C< K>< +,- / 1[\ ]( &0 % -
# CN KC< C< K>< ⇒ CN K><
< =>< &M >< *=> # T ) =>< U =< C < ? <C *C=
gA7 ( >< > >= < ∆ &0 ∆
< C>
-∠>8 ∠>E+5%>C @0 T ) =><4 > >= + a @A71 B
567∆ ∆ + -7 ∠C8 ∠CE C C= Oe f < C @0
C> ( ∠< C *∠C8+84
∠CE@0 0 C>< ( ∠CE*∠><C +E4 5%∠C8 ∠CE $ +84 &0 +E4 -7 ∠< C *∠><C # <C &%∠ *∠<
( <C *C 7<C *C=
< =>< +=< *=>4 ∠= _I1[\ = +1[\ = ⊥><4 &0 - -7 =O +1[\ =O ,- - B O ><4 < ?
(9)# ( =O @A71 B C O @0 - B =CI3h30
? 1[c ∆ =∆ + -7 ∠>=O ∠ C +84
=> C<
# ∆ =< *C< =< *=> + 50 => C< +i (
L( ∠C * ∠O=< +E4
#$ +84 &0 +E4 -7 ∠>=O *∠O=<
=< *=> ⇒ < * > + -P ^ >< = @0 _&0 O> O<4 -7 >O *> 567 / B > &0 O
< OL O *OLIOC @0 1[\ - -7 -P L # ( OC @A71 B N C @0 - B ON
< ? ∠ON *∠NO
∆ =
∆ +
CL C CO CN
(10)O0 O *OL ⇒ O *N
# ON IO 12 &M ∠ON N 12 &M ∠NO
C ∠ON *∠NO
! < =>< +∠= J!!4 &'= &-R &M >< + -P ><4 < ? = ;>< *=> ;=<
# ( >< @A71 B C >C => # ( =< @A71 B N =N = +5%=> K>< ( C / > &0 <I = K=< ( N / = &0 <4 # =>C ) 1D > +5%>C =>4
⇒ ∠>=C ∠>C=
⇒ # ∠>=C ;∠C=N ∠>=C ;∠ =C J!! C ∠C=N ∠ =C
FG =C &0 N=C = =N
∠ =C ∠C=N =C
-C ∆ =∆ + ⇒ ∠= C ∠=NC
0 ∠= C J!! ( ∠=NC J!!
# CN ⊥ =< ⇒ C< *N< +,- / 1[\ ]( &0 1[\ &-R C = ;>C ;C< *=N ;=> ;N< => ;=<
567= ;>< *=> ;=<
(11)&M >< U =< N < ? >N * +CN ;><4
5'> ⊥CN + ∈ CN4INL ⊥ >< +L ⊥ ><4 FG ∆ >N +> N J!!4 &0 ∆LN> +NL> J!!4
>N - I
∠ >N ∠LN> +&%CN ii><4
C ∆ =∆ + -7W : X
-7 > NL
Oe f ∠ >C ;∠C>< ∠ >< J!! ∠LN< ;∠N<L J!!+∆LN< L J!!4
0∠C>< ∠N<L +∆=>< ) 1D =4 -7 >C LN<
FG ∆ >C &0 ∆LN<
∠C > ∠<LN + J!!4 > NL + a i (
∠L>C ∠LN< +i (
C ∆ =∆ + ⇒ C L<
O0 > ⊥CN -7 >N * N +,- / 1[\ ]( &0 1[\ &-R C >N ;>j * N ;O>
O0 N ;>L CN ; C ;>L CN ;L< ;>L CN ;>< L( >N ;>N *CN ;><
⇒ E>N *>< ;CN⇒ >N * +CN ;><4
(12)V= ⊥ ><
: L -C _ %=C K=< &%= K=< +1[\ &-R Y Z 1[\ ](
L -C f R _
k b l C / &0 <I C K <
-7 =C K=< + % - Y Z %1[\ ]( Y Z 567=C Y Z ( =><
# < =>< < =
< ? =>< @0 &-R = &0 m >I <
V1[\ = < ? ∠> ∠ =< < ∠>=
= =
+ +
+ + = =
= + T3n m A 3o7 D -4
567 = ⇒ = ( =>< @0 &-R =
5%= ⊥>< ( ∠ 8& -7 ∠> ;∠>= 8&
5%∠>= ;∠ =< 8& -7 ∠> ∠ =< +E T n -4 $ < =>< => H =<
(13)# a A 1^ ? => : =< K>< K=> ;=<
⇒ H: K>< KH;8 ⇒ "K>< K
C 1P30 >< -7( + ( >< H %
#m -& P ) H &0 J
< =>< B C / > &0 < < ? =C Y Z l -& =><
< H f R B@0 ( &% - H @0 ( % 7@M Z p f +J*H;H4 &M A 1^ ?
567 H @0 b o JKJ;H < -& @0 H;J;J EE
FG =>C =C K=> ;>C +84
FG =<C =C K=< ;C< +E4 <P $ & +84 &0 +E4
E=C K=> ;=< ;+>C ;C<4 -7 =C K + +
QP30 P @0 IE
(14)kX 1P30 q @ @0 ]+ # a A 1^ ?
: E K]K ;E ? @0 K]KJ C ]@0 P r ( @V ( ] < q @
#m -& ) =>< => =< 8E
=> =< 8"
# =>< ) => =< 8E % 7@0 = # &67 - ( => % ( ><
L [ &67 => ;>< 8! K<= 8E
1 @0 W-&R@m+ P p 1P30 \ s @M Z 1P30 ? 567 -& =>< @0 => ;=< ;>< ;E 8E EJ
< B]b7 [\ cT
: L -=> @0 %=> >< 8" @0 ( : L - -& =>< ;E 8" ""
: L -=> >< @0 ( %=< 8" @0 < -& =>< @0 8";E E
< =><IO @0 - B >< < ? L -= J!! %=O ><
L -= *J!! %=O K >< L -= KJ!! %=O * ><
&' ( )*
+, * *- * / * ) 23 - 56 7*
8 * : / ; 2< = : > > / > ? ; 2< = :
(15)5' 12 O= ( @A71 B C OC O= -7 =C E=O
FG ∆O=> &0 ∆OC< O= OC
∠=O> ∠CO< +12 1D 4 O> O< +
C ∆O=> ∆OC< +
-7 => C< ∠>=O ∠<CO
# ∠>=O ∠<CO
0 ∠>=O &0 ∠<CO + @a 4
( => ii<C ⇒ ∠>=< ;∠=<C 89!! 56 3n &0 m A ( ]G ∆=>< &0 ∆<C= => <C =<
-C
∠>=< ∠=<C +∠>=< J!! ∠>=< ;∠=<C 89!!4 ( ∠=<C J!!
⇒ ∠>=< ∠=<C
⇒ >< =C ⇒ =O ><
∠>=< *∠=<C +∠>=< *J!! ∠>=< ;∠=<C 89!!4 ( ∠=<C KJ!!
⇒ ∠>=< *∠=<C ⇒ >< *=C
(16)∠>=< K∠=<C +∠>=< KJ!! ∠>=< ;∠=<C 89!!4 ( ∠=<C *J!!
⇒ ∠>=< K∠=<C ⇒ >< K=C
⇒ =O * ><
# @ L -∠= J!! %=O >< L(-∠= *J!! %=O K >< L -∠= KJ!! %=O * ><
< ? - P &-R P X "!! % &-R 12 &M l -7W
FG =>< ∠= J!! ∠> "!! <` ? =< ><
# ( >< @A71 B C <C <=
# =<C q ∠< t!!I=C =< <C # =>C ∠> "!! ∠=E "!!
(17)