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I saw a theorem (which is true):

Let $P$ and $Q$ be the centers of two tangent circles. Draw a line from a point through the intersection of the two circles and let that line intersect the circles at $B$ and $C$. Then $BP$ is parallel to $CQ$.

Proof:

Angles BAP and CAQ are vertical and hence equal. Triangles BAP and CAQ are isosceles with equal base angles at A. The other pair of the base angles are also equal. I.e., ∠ABP = ∠ ACQ. These two angles are internal to the lines BP and CQ and transversal BC. Since the internal angles are equal, the lines are parallel: PB || QC. (Source: here)

What can I say about the converse? i.e. if $PB || QC$ then the circles are tangent. Is that true?

I appreciate all help.

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  • $\begingroup$ What exactly is the converse statement? Which line are you going to draw to construct $B$ and $C$? $\endgroup$ – Arthur Apr 12 '18 at 12:45
  • $\begingroup$ @Arthur if PB||QC then the circles are tangent. Is that true? $\endgroup$ – Vee Hua Zhi Apr 12 '18 at 12:45
  • $\begingroup$ I can read (most times, at least). I know that that's what you're asking. I'm asking you to tell me what the points $B$ and $C$ are, if we're not already assuming that the circles are tangent. State the entire problem that you're interested in, because at the moment I can't really make sense of it. $\endgroup$ – Arthur Apr 12 '18 at 12:46
  • $\begingroup$ @Arthur demonstrations.wolfram.com/ATheoremAboutTwoTangentCircles $\endgroup$ – Vee Hua Zhi Apr 12 '18 at 12:53
  • $\begingroup$ There are diagrams about it here... $\endgroup$ – Vee Hua Zhi Apr 12 '18 at 12:53

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