Using a straightedge, a compass, and a knife to cut a triangular cake into pieces that reassemble into a rectangle

Question

Suppose you are using a triangular cake, a straightedge, a compass, and a knife. Using these, you are only allowed to cut triangular portions from the cake. Can you rearrange these portions (all) to form a rectangle?

My strategy worked, but only if I arrange only some of the portions, not all. Here is my solution-

Using the compass, I drew perpendicular line. Then I used the knife to cut a right-angled triangle. Now, comparing this piece of cake, I cut another piece of cake with one side of the cake congruent to one side of the other piece of cake.

Next, I did similar steps. Rearranging them, I got the boundary of the rectangle and now just required to fill the interior diamond.

Of course, I can stuff the remaining piece of cake in the diamond, but I am not sure whether all the pieces will be used. I want such strategy, but could not. Please help!

Answer 1

The figure below shows an approach. Find the midpoints of the sloping sides, drop perpendiculars to the base, and cut off the end triangles. The pink triangles at the ends move to the white triangles on top.