Construct an angle with two rays at point A. Create a point, P, interior to the angle. Now from an adjustable point, D, on one of the rays, draw another ray through P until it intersects the other ray From A. I labeled this point F. Create the triangle ADF and measure the Area and Perimeter.
Move point D to find the line DF which will give the minimum area of triangle ADF. Move point P and alter the angle and repeat this exercise. Can you predict where to place the line for different locations of P and different angles. Describe the relationship of the figure when the area is a minimum.
This may be a somewhat more difficult exploration… Repeat the operations above to find the line DF which minimizes the perimeter of triangle ADF. Describe the relationship of the figure when the perimeter is a minimum. (hint, there may be another line that makes this easier to see) This line is often called Philon's Line. You can find a solution to this problem by looking up Philon's line at
