Two points on the same side of the tree are 65 feet apart. The angles of elevation to the top of a 10.5 foot tree are 21˚19’ from one point and 16˚20’ from the other point. Find the distance from the tree to the closest point

Your diagram more or less matches my reading of the problem, but it seems inconsistent on several levels. After re-reading the problem, the diagram below is the one I think applies.

Usually, two observation points and two angles are used when the problem requires you to determine the height of the tree. Here, if you use the given angles and the distance between them, you find the tree needs to be 76.5 ft tall, and the nearest observation point is 196 ft away.

Since the height of the tree is given, the largest angle is from the closest viewing point, so the distance is easily found in one step.
  d = (10.5 ft)/tan(21°19') ≈ 26.9 ft

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For the viewing angles and tree height given, the distance between observation points is about 8.92 ft, not 65 ft.