1991 AIME Problems/Problem 14
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
![AIME 1991 Solution 14.png](https://wiki-images.artofproblemsolving.com//0/06/AIME_1991_Solution_14.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 • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |