1983 AIME Problems/Problem 15
Problem
The adjoining figure shows two intersecting chords in a circle, with on minor arc
. Suppose that the radius of the circle is
, that
, and that
is bisected by
. Suppose further that
is the only chord starting at
which is bisected by
. It follows that the sine of the minor arc
is a rational number. If this fraction is expressed as a fraction
in lowest terms, what is the product
?
[img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=794[/img]
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
Solution
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
[img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=796[/img]
Let be any fixed point on circle
and let
be a chord of circle
. The locus of midpoints
of the chord
is a circle
, with diameter
. Generally, the circle
can intersect the chord
at two points, one point, or they may not have a point of intersection. By the problem condition, however, the circle
is tangent to BC at point N.
Let M be the midpoint of the chord such that
. From right angle triangle
,
. Thus,
.
Notice that the distance equals
(Where
is the radius of circle P). Evaluating this,
. From
, we see that
Next, notice that . We can therefore apply the tangent subtraction formula to obtain ,
. It follows that
, resulting in an answer of
.