1994 AIME Problems/Problem 15
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Problem
Given a point 
on a triangular piece of paper 
consider the creases that are formed in the paper when 
and 
are folded onto 
Let us call 
a fold point of 
if these creases, which number three unless is one of the vertices, do not intersect. Suppose that 
and 
Then the area of the set of all fold points of can be written in the form 
where 
and 
are positive integers and is not divisible by the square of any prime. What is 
?
Solution
Let 
be the intersection of the perpendicular bisectors (in other words, the intersections of the creases) of 
and 
, and so forth. Then 
are, respectively, the circumcenters of 
. According to the problem statement, the circumcenters of the triangles cannot lie within the interior of the respective triangles, since they are not on the paper. It follows that 
; the locus of each of the respective conditions for 
is the region inside the (semi)circles with diameters 
.
We note that the circle with diameter 
covers the entire triangle because it is the circumcircle of 
, so it suffices to take the intersection of the circles about 
. We note that their intersection lies entirely within (the chord connecting the endpoints of the region is in fact the altitude of from 
). Thus, the area of the locus of (shaded region below) is simply the sum of two segments of the circles. If we construct the midpoints of 
and note that 
, we see that thse segments respectively cut a 
arc in the circle with radius 
and 
arc in the circle with radius 
.
 | The diagram shows outside of the grayed locus; notice that the creases [the dotted blue] intersect within the triangle, which is against the problem conditions. The area of the locus is the sum of two segments of two circles; these segments cut out  angles by simple similarity relations and angle-chasing.
|
Hence, the answer is, using the 
definition of triangle area, ![\left[\frac{\pi}{3} \cdot 18^2 - \frac{1}{2} \cdot 18^2 \sin \frac{2\pi}{3} \right] + \left[\frac{\pi}{6} \cdot \left(18\sqrt...](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/3/8/2/3820d89175d1215e4f2882d155818112fbec2166.gif)
, and 
.
See also
| 1994 AIME (Problems • Resources) | ||
| Preceded by Problem 14 | Followed by Last question | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
