I called the radius of circle https://data.artofproblemsolving.com...b49c39ad6d.gif: $\displaystyle r$. I called the intersection between circle https://data.artofproblemsolving.com...63d9131c1b.gif and https://data.artofproblemsolving.com...29b6d953ec.gif: $\displaystyle x$. I called the intersection between circle https://data.artofproblemsolving.com...63d9131c1b.gif and https://data.artofproblemsolving.com...e18289357d.gif: $\displaystyle y$. I called the intersection between circle https://data.artofproblemsolving.com...63d9131c1b.gif and https://data.artofproblemsolving.com...8db647f04f.gif: $\displaystyle Z$. The height of the triangle going through point x: $\displaystyle p$.The points $\displaystyle x$, $\displaystyle y$, and $\displaystyle z$ form equilateral triangle https://data.artofproblemsolving.com...ac0c17f8e7.gif. I found line https://data.artofproblemsolving.com...29b6d953ec.gifhttps://data.artofproblemsolving.com...b49c39ad6d.gif to be $\displaystyle 6+r$. The height of the triangle https://data.artofproblemsolving.com...ac0c17f8e7.gif would be $\displaystyle 15$, because the radius of circle https://data.artofproblemsolving.com...63d9131c1b.gif is 10, which would make the side length of the triangle $\displaystyle 10\sqrt{3}$. Triangle EPY is a right triangle. So line segment EP would be $\displaystyle \sqrt{(4+r)^2 - 75}$ . So line segment $\displaystyle XP$ would be $\displaystyle (6+r) + \sqrt{(4+r)^2 - 75}$, which should equal 15. Solving for $\displaystyle r$, i would get 70/13.Quote:

5. Equilateral triangle https://data.artofproblemsolving.com...ac0c17f8e7.gif is inscribed in circle https://data.artofproblemsolving.com...63d9131c1b.gif, which has radius https://data.artofproblemsolving.com...58cd7a87e5.gif. Circle https://data.artofproblemsolving.com...29b6d953ec.gif with radius https://data.artofproblemsolving.com...d7106e83bb.gif is internally tangent to circle https://data.artofproblemsolving.com...63d9131c1b.gif at one vertex of https://data.artofproblemsolving.com...ac0c17f8e7.gif. Circles https://data.artofproblemsolving.com...e18289357d.gif and https://data.artofproblemsolving.com...8db647f04f.gif, both with radius https://data.artofproblemsolving.com...a8359010b0.gif, are internally tangent to circle https://data.artofproblemsolving.com...63d9131c1b.gif at the other two vertices of https://data.artofproblemsolving.com...ac0c17f8e7.gif. Circles https://data.artofproblemsolving.com...29b6d953ec.gif, https://data.artofproblemsolving.com...e18289357d.gif, and https://data.artofproblemsolving.com...8db647f04f.gif are all externally tangent to circle https://data.artofproblemsolving.com...b49c39ad6d.gif, which has radius https://data.artofproblemsolving.com...bf762d109d.gif, where https://data.artofproblemsolving.com...3ac78c96e8.gif and https://data.artofproblemsolving.com...00c274bdaa.gif are relatively prime positive integers. Find https://data.artofproblemsolving.com...f37ec7fc1b.gif.

A alternate solution I came up with was to use triangle $\displaystyle AEX$.I know angle $\displaystyle XAE$ is 60 degrees. So using the cosine law, I would get: $\displaystyle (r+4)^2 = (r-4)^2 + 10^2 - 2*10*(r-4)*(\cos 60)$ which also comes out to $\displaystyle r$ = 70/13.

However my answer is wrong. The correct answer is 27/5. Could someone tell me what I am doing wrong? Thank you.

*I am sorry if i posted this in the wrong section. This is mostly geometry, but there is a bit of trig at the end.