Delaunay Triangulation - Forcing convex hull

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I was implementing Delaunay Triangulation based on (http://paulbourke.net/papers/triangulate/) under the impression that this triangulation always results convex hull [which is not :( ]. Is there anyway to force Delaunay Triangulation to produce convex hull always?

Concave DA sample image

Here is an example of when the triangulation does not produce the convex hull.