Glenn Lavoie
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- Mar 26, 2024
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Question - Proof Exercise: If two coplanar circles intersect in two points, then the line joining those points bisects a common tangent.
There is no diagram in the book, and I am assuming that the center of one circle could be inside, outside, or on the circle of the other circle. It seems that I need to prove three cases, but for now I am focused on proving the case in which one circle is larger than the other circle, and the center of the smaller circle is located outside the larger circle. I tried solving it by drawing some line segments. I figured that if I could prove that CE = EF, then I could deduce that GH = HI by proportionality, which would mean that line l bisects the common tangent (I couldn't figure out how to show that CE and EF are line segments in this text box, the usual bar over the two letters. Please see diagram. You might prefer to skip the following explanations below, describing the line segments I drew). I drew a line segment from the center of the larger circle A to the center of the smaller circle B, perpendicular to the line l that joins the points where the circles intersect, and then extended it further, such that it is a diameter of the smaller circle B. Then, I drew a radius from the center of the larger circle A to point G, where the common tangent meets circle A, and drew a radius to point I, where the common tangent meets the smaller circle B. It's apparent that these two radii are parallel. I also drew two line segments perpendicular to the line AB, which joins the two circles, one segment from point G to point C, and one from point I to point F. These two line segments, IF and GC, are parallel. The line l passes through the common tangent at point H, and HE, a segment of line l, is perpendicular to AB. I figured that if I could prove that CE = EF, then I could deduce that GH = HI by proportionality, which would prove that line l bisects the common tangent. But after at least two or three months of tinkering with this problem, I figure that it's a dead end, or I am not recalling a crucial theorem. I also tried drawing the other line segments shown in the diagram but got nowhere that way either. If I failed to communicate clearly, please let me know for next time. I had to repost this, because the diagram I attached went missing. I would greatly appreciate any pointers.
There is no diagram in the book, and I am assuming that the center of one circle could be inside, outside, or on the circle of the other circle. It seems that I need to prove three cases, but for now I am focused on proving the case in which one circle is larger than the other circle, and the center of the smaller circle is located outside the larger circle. I tried solving it by drawing some line segments. I figured that if I could prove that CE = EF, then I could deduce that GH = HI by proportionality, which would mean that line l bisects the common tangent (I couldn't figure out how to show that CE and EF are line segments in this text box, the usual bar over the two letters. Please see diagram. You might prefer to skip the following explanations below, describing the line segments I drew). I drew a line segment from the center of the larger circle A to the center of the smaller circle B, perpendicular to the line l that joins the points where the circles intersect, and then extended it further, such that it is a diameter of the smaller circle B. Then, I drew a radius from the center of the larger circle A to point G, where the common tangent meets circle A, and drew a radius to point I, where the common tangent meets the smaller circle B. It's apparent that these two radii are parallel. I also drew two line segments perpendicular to the line AB, which joins the two circles, one segment from point G to point C, and one from point I to point F. These two line segments, IF and GC, are parallel. The line l passes through the common tangent at point H, and HE, a segment of line l, is perpendicular to AB. I figured that if I could prove that CE = EF, then I could deduce that GH = HI by proportionality, which would prove that line l bisects the common tangent. But after at least two or three months of tinkering with this problem, I figure that it's a dead end, or I am not recalling a crucial theorem. I also tried drawing the other line segments shown in the diagram but got nowhere that way either. If I failed to communicate clearly, please let me know for next time. I had to repost this, because the diagram I attached went missing. I would greatly appreciate any pointers.