Geometry: An Interactive Journey to Mastery

4. 6XSSRVHWKDWWKHJLYHQFLUFOHKDVUDGLXVr. Call its center F.
   'UDZDOLQHSDUDOOHOWRWKHJLYHQOLQHrXQLWVEHORZLW
   Call this line m.
   The diagram in Figure S.29.3 shows that the center of
   any circle tangent to the given circle and the given line
   is equidistant from F and m7KH\WKXVOLHRQDSDUDEROD
   with focus FDQGGLUHFWUL[m.

Lesson 30

1. 6XSSRVHWKDWSRLQWVA and BDUHUHÀHFWHGDERXWDOLQHL to the
   points Ac and Bc, respectively. First assume that A and B lie on the
   same side of L.
   $OVROHWPEHWKHSRLQWRILQWHUVHFWLRQRIAAc and line L and Q
   EHWKHLQWHUVHFWLRQRIBBc and L௘6HHFigure S.30.1௘ͽ
   Now, +BQP and +BQPc DUHFRQJUXHQWE\6$6ͼ௘BQ B Qc ,ERWK
   KDYHDULJKWDQJOHDQGERWKVKDUHPQ௘ͽ7KXVWKHWZRDQJOHVODEHOHGx
   DUHFRQJUXHQWͼ௘EHFDXVHWKH\DUHPDWFKLQJDQJOHVLQFRQJUXHQWWULDQJOHV௘ͽ
   and PB PBc,ͼ௘EHFDXVHWKH\DUHPDWFKLQJVLGHVLQFRQJUXHQWWULDQJOHV௘ͽ
   Thus, ‘#‘APB A PBcc,EHFDXVHWKH\ERWKKDYHPHDVXUHíx.
   6R+APB is congruent to +APBccE\6$6ͼ௘AP = AƍP, ‘#‘APB A PBcc, and PB = PBƍ௘ͽ
   Thus, AB = AƍBƍͼ௘EHFDXVHWKH\DUHPDWFKLQJVLGHVLQFRQJUXHQWWULDQJOHV௘ͽ
   6RWKHUHÀHFWLRQKDVLQGHHGSUHVHUYHGWKLVGLVWDQFHEHWZHHQA and B.
   In the same way, you can show that this distance is preserved in the case where A and B lie on opposite
   sides of the line, in the case where one of the points A or BOLHVRQWKHOLQHDQGLQWKHFDVHZKHUHWKH\ERWK
   GRͼ௘&KHFNWKLV௘ͽ
   7KXVDUHÀHFWLRQSUHVHUYHVWKHGLVWDQFHVEHWZHHQDOOSDLUVRISRLQWVLQWKHSODQHDQGWKHUHIRUH
   is an isometry.

rr

rr

F

m

Figure S.29.3

A

xx

Aމ

B

Q

P L

Bމ

Figure S.30.1