:    . 12 
 


       ?        ,          .   12  , 24  , ,    36  . : ,  , ,  , , ,     .  ,           .          :        .





 

:    . 12 





  





      


  ,   ,  ,   ,       .          .       :      ,       .

           .      ,   .      ;      ,    .   ,         .




   


    : ,  ,   .      .     ,   ,   .          :  , ,      .

,   M   AB   M  AB   AM = MB.  CM  AB   :       C.          .




    


AB      A  B,        .   ?ABC      B.  AB ? CD   ,  AB ? CD   .  ~   ;       .        .

     ,        .      .  ,   , ,         .    , ,       .

         .      ,     ,   .     :          .




    


     ,          .     :  ,   ,      .    .          .

     .      :    ,    ,    .   .     ,      . ,              .




  


        .       ,         .           ,   ,   ,  .

     . ,        ,             .      ,      ,       .




  


1.  ,     (#case-1-start)

2.      (#case-2-start)

3.      (#case-3-start)


4.     (#case-4-start)

5.     (#litres_trial_promo)

6.     (#litres_trial_promo)

7.      (#litres_trial_promo)

8.    ,   (#litres_trial_promo)

9.       (#litres_trial_promo)

10.       (#litres_trial_promo)

11.       (#litres_trial_promo)

12.       (#litres_trial_promo)




   





 1.  ,    


:             .



 

   180;      .

    ,      ,     .

    ,       .


  

        .   ?AOC   O,   OA  OC.   AB  CD   O      :      AOB  COD.  ,  OA  OB,   OC  OD   .           .

      .  AOC  COB    180.  BOD  COB   180,  OC  OD .    COB,  ?AOC = ?BOD.  ?AOD = ?COB.         ,      .

        .          .        ,     ,     .         : , , ,  .


 

   AB  CD   O.  O      AB  CD. : ?AOC = 2x + 15, ?COB = 4x ? 15.         .

:  x      ;    .




 ;      .




1.  AOB ,   OA  OB .   AOC  COB   OC,    ,   .

2.     180: (2x + 15) + (4x ? 15) = 180.      6x = 180,  x = 30.

3.       : ?AOC = 2  30 + 15 = 75, ?COB = 4  30 ? 15 = 105.     0  180,     .

4.  COD  ?COB + ?BOD = 180.  ?BOD = 180 ? 105 = 75.          .

5.  AOD  AOC  :    OA,   OD  OC . , ?AOD = 180 ? 75 = 105.

6.   : 75 + 105 + 75 + 105 = 360.     ,          .

: x = 30; ?AOC = ?BOD = 75, ?COB = ?AOD = 105.

       .       .         ;        .


  

 :    , ,    90.

    .        ;  ,  75  105        90.

 :       .       ,      .


 1.1

  AB  CD   O; O   AB  CD.  ?AOD  ?DOB   1 : 3.

:           .




   ,     .

   1.1 (#litres_trial_promo)

  1.1 (#litres_trial_promo)


 1.2

 AB  CD     O; O   AB  CD. ,  ?AOC = ?COB.  OM     BOD  ?BOM = 30.

:  AB ? CD   ?MOD.




       .

   1.2 (#litres_trial_promo)

  1.2 (#litres_trial_promo)


    

      ,   .          ;       .         ,        .

  (#navigation-start)




 2.     


:           ,    .



 

      ,  ;       .

   180;    ,      .

         .

              .


  

        .   A  ,  BC.        AB, AC   B  C    .    BAC      A.  ?A + ?B + ?C = 180    .

   C    ACD,  B, C, D        .    CA   BC  C.  ACB  ACD .    180    ,  ?ACD = ?BAC + ?ABC.      ,     .

       .      ,       .        ,         .        ,    .


 

ABC   .  D    BC  C,   BCD.  ?BAC = 2t + 10, ?ABC = t + 20, ?ACD = 5t ? 10  AB = 13.  t  .

:  t    ,     . ,    AB.




MN      A; D   C.




1.  A  MN ? BC.  M  N     A ,   AM    CB,  AN   BC.        BC;  AB  AC      MN.

2.   AB  ?MAB = ?ABC,    AC  ?CAN = ?ACB:       .

3.  MAB, BAC  CAN     MAN.     ?ABC + ?BAC + ?ACB = 180.     .

4.   BCD  ?ACB + ?ACD = 180.            ACB,  ?ACD = ?BAC + ?ABC.

5.  : 5t ? 10 = (2t + 10) + (t + 20). , 2t = 40  t = 20.

6.  ?A = 50, ?B = 40, ?ACD = 90.  ?C = 180 ? 90 = 90.         180.

7.  AB = 13       .     ,      ;        .

: t = 20; ?A = 50, ?B = 40, ?C = 90.  AB    .

          .       .   BC      ,   .


  

 :        .

     180,      180:      .          .

 :            .      :       .


 2.1

   ABC  CA   A  D,   CB   B  N.  : CAD  CBN.    DAB  ABN  250.

:  ?ACB,      A  B.




   ;     .

   2.1 (#litres_trial_promo)

  2.1 (#litres_trial_promo)


 2.2

   ABC  B  C  45  75.    A   BC  M,  BMC.

:  ?AMC  ?AMB    .




AM   A;  M   BC.

   2.2 (#litres_trial_promo)

  2.2 (#litres_trial_promo)


    

        180     .           .        ;     ,     .

  (#navigation-start)




 3.     


:             .



 

     :      .

         ;     .

     180;     .

            .


  

  ,     .  AB = AC,   BC,      BAC.       :  AB   C,  AC   B.     :            .

       ,       .     :   ,      .         .        .

      :          .         .                .         .


 

ABC   , AB = AC  ?BAC = 44.  M    BC; , BMC  BM = MC.

:             .




M   ;  AM     .




1.   ABM  ACM.  : A    BC,  M   B  C.    A ? A, B ? C, M ? M.

2.   AB = AC.    BM = CM. , AM = AM,          .

3.     .   ABM  ACM    .     M,     A     .

4.     ?ABM = ?ACM.  M   BC,  BM   BC,  CM   CB. , ?ABC = ?ACB.           AB = AC.

5.       ?.     44 + 2? = 180.  2? = 136  ? = 68.

:      ?B = ?C;  ?A = 44      68.

  ,    .       ,       .      ,     .


  

 :   ,     .

      .  AB = AC   B  C,   A     .   44   68.

 :   ,    .        ,      ,      .


 3.1

   ABC  ?ABC = ?ACB.    A  BC  D,  BDC.

:   : AB = AC.        .




     ;     .

   3.1 (#litres_trial_promo)

  3.1 (#litres_trial_promo)


 3.2

   ABC  AB = AC  ?BAC = 36.   AB   D   A  B,  CD = CB.

:  ?ACD  ?BCD,   AD = CD.




D   AB;    CD  CB.

   3.2 (#litres_trial_promo)

  3.2 (#litres_trial_promo)


    

    :      ,         .          .              .

  (#navigation-start)




 4.    


:   ,   ,        .



 

             .

    90;   .

        ;       .

 :         .


  

      .    .        .    ,     .      AM   BM = MC   M  BC;        A  M.

              .           .       :    A  ,      M  .

    .    ,          ,           .    ,        :         .           .


 

ABC   , AB = AC.  AM     BC,   M   BC  BM = MC.

: ,  AM        BAC.




  .


   .

   ,     (https://www.litres.ru/book/raznoe/geometriia-ot-usloviia-k-dokazatel-stvu-12-razborov-74418009/)  .

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