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how to prove that any connected 2 circles by 1 point have a straight line between centers through connection point

enter image description here

How to prove that the line xy is straight line or the angle between the lines x and y is 180 degrees?

I got answers assuming the 2 circle should intersect in 2 point.

my question is the 2 circles are connected in 1 point only

so assuming there will be triangle then the pic will look like this

enter image description here

ac & ad are the radius of the small circle

bc & be are the radius of the big circle

how to prove that ab is a straight line, in another way other than the perpendicular tangent way

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  • $\begingroup$ Suppose the circle centers are $A,B$ and the point of tangency is $C$. If $C$ does not lie on the line $AB$, then $ABC$ is a triangle. Can you construct another triangle $ABD$ with $AC=AD,BC=BD$? What can you say about $D$ in relation to the circles? $\endgroup$ – Rahul Oct 6 '18 at 4:51
  • $\begingroup$ @Rahul are you assuming D is another tangency point? $\endgroup$ – asmgx Oct 6 '18 at 7:02
  • $\begingroup$ I'm not assuming anything. I'm showing how you can prove that if $xy$ is not a straight line, the circles cannot meet at only one point. $\endgroup$ – Rahul Oct 6 '18 at 16:08
  • $\begingroup$ @Rahul i edited the question $\endgroup$ – asmgx Oct 7 '18 at 2:03
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Notice that the radius allways form a straight angle with the tangent, in particular every tangent line touches one and only one point of the circle. If the circles are touching only one point of each other, the tangent line that touches that point is the same for both circles. Take now the common point and make a straight line from that point to te center of every circle. As the tangent makes a $\pi/2$ angle with each radius, then the angle between both radius has to be $\pi$

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  • $\begingroup$ is there any other way to prove it without the tangent line $\endgroup$ – asmgx Oct 6 '18 at 9:03
  • $\begingroup$ shure, lets supose that the angle between both radius is less than $\pi$, then you can make a triangle with the two radius common point, and the circle's centers, notice that the 3rd line you made is never outside the circles, (maybe you can prove this), so using that the circles are touching themselves in 2 points, and that contradicts the hypotesis $\endgroup$ – Pablo Valentin Cortes Castillo Oct 6 '18 at 15:39
  • $\begingroup$ I edited the question $\endgroup$ – asmgx Oct 7 '18 at 2:03

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