Minimizing the Area of a Triangle

Suppose a segment is drawn in the first quadrant of the coordinate plane and has variable slope. Suppose this segment passes through the point (2,3).   This line also has an x-intercept of (c,0) and a y-intercept of (0,d), where c, d > 0.   Use calculus to algebraically determine the slope of this line for which the area of the displayed right triangle is minimum. Then use calculus to prove this area is indeed the minimum area. How does your result compare with what this applet suggests? Note: The red point is moveable.
Note to Instructors/Students: The purple point is moveable too, should you wish to solve a problem with similar context.

Quick (Silent) Demo