Interesting problem! Do you have any ideas?How to construct a rectangle given one of his sides(a) and the diference between the diagonal and the other side(b)?
The construction will be totally geometric. Only the thinking to discover it will be algebraic.I was thinking the same way. An idea for a more geometric solution maybe?
Thank you for your help. I would like to share another idea. If the rectangle is ABCD, AB=a,BC=b and BD=d, than we can prolong the side DA through the point A to a point E so AE=d-b. Now we can construct the triangle ABE given (side a,side d-b and right angle). Considering the fact that the triangle EBD is isosceles triangle( DE=d and BD=d) than the bisector of BE should pass through the point D. We get both sides of the rectangle and we can easily find the point C.The construction will be totally geometric. Only the thinking to discover it will be algebraic.
Are you saying you want to tie one hand behind your back by forcing yourself to do all your thinking geometrically, as Euclid might? I suppose it's possible, but I have no interest in trying.
Nice method; and it can indeed be discovered just by drawing the final figure and working backward, trying to get to a point that could be constructed using only what's given.Thank you for your help. I would like to share another idea. If the rectangle is ABCD, AB=a,BC=b and BD=d, than we can prolong the side DA through the point A to a point E so AE=d-b. Now we can construct the triangle ABE given (side a,side d-b and right angle). Considering the fact that the triangle EBD is isosceles triangle( DE=d and BD=d) than the bisector of BE should pass through the point D. We get both sides of the rectangle and we can easily find the point C.
We can use the similar thinking if we have to construct a rectangle given a and d+b.