+0 In triangle $STU$, let $M$ be the midpoint of $\overline{ST},$ and let $N$ be on $\overline{TU}$ such that $\overline{SN}$ is an altitude of triangle $STU$. If $ST = 8$, $SU = 8$, $TU = 8$, and $\overline{SN}$ and $\overline{UM}$ intersect at $X$, then what is $SX$?
+129850 
This one is pretty easy
The triangle is equilateral
The altitude of an equilateral triangle is a median
And since M is the midpoint of ST, then UM is also a median
Their intersection, X, is the centroid of STU
SN = sqrt [ST^2 - TN^2] = sqrt [ 8^2 - 4^2] = sqrt [ 48] = 4sqrt (3)
By a property of centroids ....SX = (2/3)(SN)
So
SX = (2/3) * 4 (sqrt3) = 8sqrt (3) / 3 = 8 / sqrt 3 ≈ 4.618
