1991 AIME Problems/Problem 14
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Problem
A hexagon is inscribed in a circle. Five of the sides have length
and the sixth, denoted by
, has length
. Find the sum of the lengths of the three diagonals that can be drawn from
.
Solution
![Click to view code [Asy_image]](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/2/9/2/292cf3feb43ab5c4382ea5661c3100c4014e0ff3.png)
Let
,
, and
.
Ptolemy's Theorem on
gives
, and Ptolemy on
gives
.
Subtracting these equations give
, and from this
. Ptolemy on
gives
, and from this
. Finally, plugging back into the first equation gives
, so
.
See also
| 1991 AIME (Problems • Resources) | ||
| Preceded by Problem 13 | Followed by Problem 15 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||




