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Start by taking any 3 points and make a triangle (let lines extend outside the triangle), you then end up with one of 6 cases, which the following pictures show.

Case 1: There is a 4th point in a "side area". Picture show solution.

Case 2: There is a 4th point in a "corner area" and a 5th point in another "corner area". Picture shows solution.

Case 3: Both point 4 and point 5 are in the same corner area, and when lines from the two other corners are folded towards them, each line finds its own point. Picture shows solution.

Case 4: Same as case 3, except that when the lines from the two other corners are folded towards them, both lines finds the same point. Then draw a line between that point and the close corner, the 5th point ends up on either side of that line. Picture show solution.

Case 5: A 4th point is in a corner area and the 5th point is inside the triangle. Draw a line from the 4th point through the close corner and let it extend into the triangle. The 5th point ends up on either side of that line. Picture show solution.

Case 6: The 4th and 5th point are both inside the triangle. Draw lines from each corner through one of the points. Depending on where the other point ends up make the convex quadrilateral according to the picture.