2011 AIME I Problem 8

In triangle ABC, BC = 23, CA = 27, and AB = 30. Points V and W are on \overline{AC} with V on \overline{AW}, points X and Y are on \overline{BC} with X on \overline{CY}, and points Z and U are on \overline{AB} with Z on \overline{BU}. In addition, the points are positioned so that \overline{UV} \parallel \overline{BC}, \overline{WX} \parallel \overline{AB}, and \overline{YZ} \parallel \overline{CA}. Right angle folds are then made along \overline{UV}, \overline{WX}, and \overline{YZ}. The resulting figure is placed on a level floor to make a table with triangular legs. Let h be the maximum possible height of a table constructed from triangle ABC whose top is parallel to the floor. Then h can be written in the form \frac{k \sqrt{m}}{n}, where k and n are relatively prime positive integers and m is a positive integer that is not divisible by the square of any prime. Find k + m + n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.