Problem:
A side-angle-side triangle has sides $4$ and $\frac{13}{2}$ and an angle $6 {}^{\circ}$. Determine the centroid, circumcenter, incenter, orthocenter, nine point center, and symmedian point of the triangle.
Answer:
NinePointCenter: $\left\{\frac{571-416 \cos (6 {}^{\circ})}{8 \sqrt{233-208 \cos (6 {}^{\circ})}},-\frac{(233 \cos (6 {}^{\circ})-208) \csc (6 {}^{\circ})}{8 \sqrt{233-208 \cos (6 {}^{\circ})}}\right\}$ 
SymmedianPoint: $\left\{\frac{13 \sqrt{233-208 \cos (6 {}^{\circ})} (4 \cos (6 {}^{\circ})-13)}{208 \cos (6 {}^{\circ})-466},\frac{26 \sin (6 {}^{\circ}) (233-208 \cos (6 {}^{\circ}))^{3/2}}{-72696 \cos (6 {}^{\circ})+10816 \cos (12 {}^{\circ})+65105}\right\}$ 
Circumcenter: $\left\{\frac{1}{4} \sqrt{233-208 \cos (6 {}^{\circ})},\frac{1}{4} \sqrt{233-208 \cos (6 {}^{\circ})} \cot (6 {}^{\circ})\right\}$ 
Orthocenter: $\left\{-\frac{13 (8 \cos (6 {}^{\circ})-13)}{2 \sqrt{233-208 \cos (6 {}^{\circ})}},\frac{(8 \cos (6 {}^{\circ})-13) (13 \cos (6 {}^{\circ})-8) \csc (6 {}^{\circ})}{2 \sqrt{233-208 \cos (6 {}^{\circ})}}\right\}$ 
Centroid: $\left\{\frac{67-52 \cos (6 {}^{\circ})}{\sqrt{233-208 \cos (6 {}^{\circ})}},\frac{52 \sin (6 {}^{\circ})}{3 \sqrt{233-208 \cos (6 {}^{\circ})}}\right\}$ 
Incenter: $\left\{\frac{13 \left(\sqrt{233-208 \cos (6 {}^{\circ})}-8 \cos (6 {}^{\circ})+13\right)}{2 \left(\sqrt{233-208 \cos (6 {}^{\circ})}+21\right)},\frac{52 \sin (6 {}^{\circ})}{\sqrt{233-208 \cos (6 {}^{\circ})}+21}\right\}$
{'source': 'AMPS', 'file': 'amps/mathematica/geometry/triangles/16241.txt'}