Two circles intersect at C and D, and their common tangents intersect at T. CP and CQ are the tangents at C to the two circles; prove that CT bisects ∠ P C Q.

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Two circles intersect at C and D, and their common tangents intersect at T. CP and CQ are the tangents at C to the two circles; prove that CT bisects   ∠  P  C  Q.

Sarah Sutton · Accepted Answer

Step 1Let       Γ    1   and       Γ    2   be our circles,       Γ    1    ∩      Γ    2    =  {  C  ,  D  }, TM and TK be common tangents to our circles, where   {  M  ,  K  }  ⊂      Γ    1    ..Also, let P and Q be placed on segments MT and KT respectively such thatCP and CQ are tangents to       Γ    2   and       Γ    1   respectively.Now, let   T  C  ∩      Γ    1    =  {  C  ,  N  } and let f be a homothety with the center T such that   f  (      Γ    2    )  =      Γ    1  .Step 2Thus,   f  (  {  C  }  )  =  {  N  } and f(CP) is a tangent to the circle       Γ    1   in the point N, which is parallel to CP and let this tangent intersects CQ in the point R.Let   C  P  ∩      Γ    1    =  {  L  ,  C  }.Thus, since NR||LP, we obtain   ∡  P  C  T  =  ∡  N  C  L  =  ∡  R  N  C  =  ∡  R  C  N  =  ∡  Q  C  T and we are done

Two circles intersect at C and D, and their common tangents intersect at T. CP and CQ are the tangents at C to the two circles; prove that CT bisects angle PCQ.

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