1952 AHSME Problems/Problem 8
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Problem
Two equal circles in the same plane cannot have the following number of common tangents.
Solution
Two congruent coplanar circles will either be tangent to one another (resulting in 
common tangents), intersect one another (resulting in 
common tangents), or be separate from one another (resulting in 
common tangents).
Having only 
common tangent is impossible, unless the circles are non-congruent and internally tangent.
![[asy] pair A=(1,0), B=(-1,0), C=(-5,0), D=(-6,0), E=(5,0), F=(8,0); draw(circle(A,1)); draw(circle(B,1)); draw(circle(C,1)); draw(circle(D,1)); draw(circle(E,1)); draw(circle(F,1)); draw((0,1)--(0,-1),red); draw((-2,1)--(2,1),red); draw((-2,-1)--(2,-1),red); draw((-7,1)--(-4,1),red); draw((-7,-1)--(-4,-1),red); draw((4,1)--(9,1),red); draw((4,-1)--(9,-1),red); draw((5+0.5sqrt(2),0.5sqrt(2))--(8-0.5sqrt(2),-0.5sqrt(2)),red); draw((5+0.5sqrt(2),-0.5sqrt(2))--(8-0.5sqrt(2),0.5sqrt(2)),red); label("$3$",(-1.5,0),E); label("$1$",(0,0.75),N); label("$2$",(0,-0.75),S); label("$1$",(-5.5,0.75),N); label("$2$",(-5.5,-0.75),S); label("$1$",(6.5,0.75),N); label("$2$",(6.5,-0.75),S); label("$3$",(6.5,-0.25),N); label("$4$",(6.5,0.25),S); [/asy]](http://latex.artofproblemsolving.com/8/d/f/8df173732e90d05a18a107a1a8ee0e0a2ce00311.png)
See also
| 1952 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
