Triangle to Conic?

Creation of this applet was inspired a problem posted on Alexander Bogomolny's website: Cut-The-Knot.org. Suppose P is a point in the interior of any triangle. Draw lines through P that each line is parallel to a side of the triangle. (There are 3 such lines.) The 6 points at which these lines meet the triangle's sides WILL ALWAYS lie on a conic section. Try it! Slide the slider and then move the purple point around. Do you get different types of conic sections? (You can use the blue slider to adjust the size of the blue angle.) How can you formally prove what this applet informally illustrates?