Math question that bugs me to death

Discus: SAT/ACT Tests and Test Preparation: July 2004 Archive: Math question that bugs me to death
 By Legendofmax (Legendofmax) on Saturday, July 10, 2004 - 09:15 pm: Edit

If three different circles are drawn on a piece of paper, at most how many points can be common to all three?

 By Texas137 (Texas137) on Saturday, July 10, 2004 - 09:53 pm: Edit

you can make a region as large as the circles common to all three, so I would say it's an infinite number of common points. If you mean points of intersection, I would say just one.

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 By Devious (Devious) on Saturday, July 10, 2004 - 09:55 pm: Edit

If they're different, then I'd say 6?

Edit: I just realised that the question was asking for points common to all three circle, in that case, I think they're 2.

 By Benzinspeicher (Benzinspeicher) on Saturday, July 10, 2004 - 09:56 pm: Edit

i think it'd be 2 points. Since 2 circles have only points of intersection, the 3rd circle can have its center right in the middle of the distance from the two points of intersection. Need backup on this.

 By Benzinspeicher (Benzinspeicher) on Saturday, July 10, 2004 - 10:02 pm: Edit

oops, i meant to say 2 circles have only 2 points of intersection at max.

 By Feuler (Feuler) on Saturday, July 10, 2004 - 10:11 pm: Edit

The max is two. If you pick any two points A and B, any circle drawn with center on a point equidistant from A and B with appropriate radius will pass through A and B, thus there are any infinite number of such points (their locus is the perpindicular bisector of segment AB).

Intuitively, the answer cannot be more that two; more rigorously, this is because any three non-colinear points have a unique circle passing through all three (because the center must be equidistant from A and B as well as B and C, it is defined as the intersection of two lines, namely the perp. bisectors of AB and BC), thus no more than one circle can pass through each of three points.

 By Tongos (Tongos) on Saturday, July 10, 2004 - 10:17 pm: Edit

solve x^x=5 what is x, hehehehehe. ah, ha, ha ,ha ,ha

 By Legendofmax (Legendofmax) on Saturday, July 10, 2004 - 11:53 pm: Edit

I guess I do not understand... if I draw three circles with a common overlap then wouldn't that central area encompass three intersections?

 By Benzinspeicher (Benzinspeicher) on Sunday, July 11, 2004 - 12:22 am: Edit

start off by drawing two circles, and they can only intersect at one point (tangent), or two points-therefore, it cannot be more than two-just set the third circle's center to be exactly half way along the distance from the two points of intersection of the first two circles

 By Needhelp06 (Needhelp06) on Sunday, July 11, 2004 - 01:01 am: Edit

its 2 points