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.

Liberty Page 2022-09-17 Answered
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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Answers (1)

Sarah Sutton
Answered 2022-09-18 Author has 4 answers
Step 1
Let 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 that
CP 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 2
Thus, 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

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