Geometry 2D

Tangent lines to circles

The equations of the following tangents to one or two circles are calculated. The constructions are drawn and the construction lines are shown if necessary.

Example 1:

Given: 
¯¯¯¯¯
  k: M(5|8), r=8.48528, B(-1|2)              

Tangent to k in B 
¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  t: x + y = 1

Example 2:

Given:
¯¯¯¯¯
k : M(5|8) ,   r=5 ,    P(-1|2)

Tangents to k through P
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  t1: 1,06014·x − 6,77319·y = -14,6065
  t2: 6,77319·x − 1,06014·y = -8,89348

Points of contact
¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  k n t1 = B1(5,77319|3,06014)
  k n t2 = B2(0,0601439|8,77319)

Example 3:

Given:
¯¯¯¯¯
  k : M(5|8) ,   r=5 ,      g : -x + 3·y = 0  

Tangents on k parallel to g
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  t1: x − 3·y = -3,18861
  t2: x − 3·y = -34,8114

Points of contact
¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  k n t1 = B1(6,58114|3,25658)
  k n t2 = B2(3,41886|12,7434)

Example 4:

Given:
¯¯¯¯¯
  k1 : M(5|8) ,    r=5
  k2 : M(-1|2) ,   r=3

Outer tangents
¯¯¯¯¯¯¯¯¯¯¯¯
  t1: -4,2923·x + 7,04104·y = -6,36427
  t2: -7,04104·x + 4,29230·y = 40,3643

Inner tangents
¯¯¯¯¯¯¯¯¯¯¯¯
  t3: 1,21895·x + 2,55228·y = 12,3709
  t4: -2,55228·x − 1,21895·y = -8,3709
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