Difference between revisions of "2020 CIME I Problems/Problem 14"
(Created page with "==Problem 14== Let <math>ABC</math> be a triangle with sides <math>AB = 5, BC = 7, CA = 8</math>. Denote by <math>O</math> and <math>I</math> the circumcenter and incenter of...") |
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==Solution== | ==Solution== | ||
| − | Analytic geometry gives us <cmath>DK=\frac{17 \ | + | Analytic geometry gives us <cmath>DK=\frac{17\sqrt57}{19}.</cmath> The answer is <math>93</math>. |
==See also== | ==See also== | ||
Revision as of 11:55, 1 September 2020
Problem 14
Let 
be a triangle with sides 
. Denote by 
and 
the circumcenter and incenter of 
, respectively. The incircle of touches 
at 
, and line 
intersects the circumcircle of 
again at 
. Then the length of 
can be expressed in the form 
, where 
are positive integers, 
and 
are relatively prime, and 
is not divisible by the square of any prime. Find 
.
Solution
Analytic geometry gives us ![\[DK=\frac{17\sqrt57}{19}.\]](http://latex.artofproblemsolving.com/4/0/f/40f5721a5a03e7c64ae2ce3134276a16a4bae9d3.png)
The answer is 
.
See also
| 2020 CIME I (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 CIME Problems and Solutions | ||
The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions. 
