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Two non-intersecting circles have 4 common tangents (two external, two internal). In a sentence with fewer than 8 words, explain why the midpoints of these tangents lie on the same line.

By Zerg_Vvins (Zerg_Vvins) on Friday, August 15, 2003 - 01:41 pm: Edit

holes = midpoints lines=tangents

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Is that it? probably wrong though lol.

By Serene (Serene) on Friday, August 15, 2003 - 01:47 pm: Edit

...... zerg, did you draw it out? what 'holes'?

By Zerg_Vvins (Zerg_Vvins) on Friday, August 15, 2003 - 01:49 pm: Edit

yeah, the only part I dont get is "(two external, two internal)" all my tangents seems to be external

By Jason817 (Jason817) on Friday, August 15, 2003 - 01:49 pm: Edit

I don't even understand how 2 non-intersecting circles can even have 4 common tangents.

By Serene (Serene) on Friday, August 15, 2003 - 01:50 pm: Edit

internal as in, crossing the space right between the two centers i guess

By Zerg_Vvins (Zerg_Vvins) on Friday, August 15, 2003 - 01:54 pm: Edit

hum..

well I guess then my picture should be the solution. Just imagine two circles on either side of it and you will see it.

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Two circles on either side of it and the lines I have drawn should be the tangents and the midpoints of these tangents are shown as holes in my drawing

By Serene (Serene) on Friday, August 15, 2003 - 01:55 pm: Edit

zerg: the two circles don't have to be same size.

By Zerg_Vvins (Zerg_Vvins) on Friday, August 15, 2003 - 01:56 pm: Edit

easier that way
the problem didn't explicitly forbid same size cirlces

By Fairyofwind (Fairyofwind) on Friday, August 15, 2003 - 01:57 pm: Edit

No, you can't go without loss of generality the two circles are the same size.

By Serene (Serene) on Friday, August 15, 2003 - 01:58 pm: Edit

zerg: stop trying to worm your way out of a problem. because the problem didn't specify sizes, just one particular case cannot serve as the solution.

By Zerg_Vvins (Zerg_Vvins) on Friday, August 15, 2003 - 01:59 pm: Edit

oh yeah if not then the line drawn will be in an arch huh

By Zerg_Vvins (Zerg_Vvins) on Friday, August 15, 2003 - 02:00 pm: Edit

Serene: What are you talking about?
Isn't that the solution?

By Fairyofwind (Fairyofwind) on Friday, August 15, 2003 - 02:01 pm: Edit

If the line drawn was an arch, it isn't a line...

By Serene (Serene) on Friday, August 15, 2003 - 02:02 pm: Edit

fairy: no weird theorem right?

By Fairyofwind (Fairyofwind) on Friday, August 15, 2003 - 02:05 pm: Edit

Hint: Power of a point.

By Zerg_Vvins (Zerg_Vvins) on Friday, August 15, 2003 - 02:07 pm: Edit

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FAIRY: just put two equal sized circles on either side!!!
do you not see my "POWER OF A POINT"??

By Fairyofwind (Fairyofwind) on Friday, August 15, 2003 - 02:09 pm: Edit

Zerg: ...

By Zerg_Vvins (Zerg_Vvins) on Friday, August 15, 2003 - 02:13 pm: Edit

Ok ok Serene you figure it out, I am done...
I probably misread the question or something

By Fairyofwind (Fairyofwind) on Friday, August 15, 2003 - 02:15 pm: Edit

Zerg you didn't misread it. But you assumed the two circles have the same size. What happens if they don't?

By Serene (Serene) on Friday, August 15, 2003 - 02:21 pm: Edit

seems too few congruent angles to use power of a point though? *_*

By Xiggi (Xiggi) on Friday, August 15, 2003 - 02:25 pm: Edit

And what happens if the smaller circle is inscribed in a larger one?

Anyhow, lines intersections and uniqueness of points are items that Zerg consider open to debate

By Fairyofwind (Fairyofwind) on Friday, August 15, 2003 - 02:30 pm: Edit

Hint 2: When two circles intersect, their internal tangents "degenerate" into their points of intersection. If you wish, consider that case first.

By Serene (Serene) on Friday, August 15, 2003 - 02:35 pm: Edit

wa, Fairy is a genius ^^

ok i got that case. let me look at the original problem again =)

By Xiggi (Xiggi) on Friday, August 15, 2003 - 02:57 pm: Edit

One silly question. How do you define the midpoint of a tangent?

Or do we to assume that the midpoints are the midpoints of the segments formed by the corresponding points of tangency?

By Fairyofwind (Fairyofwind) on Friday, August 15, 2003 - 03:15 pm: Edit

That is correct.