Scaling a parallelogram to fit a triangle

Question

How do I, with straightedge and compass, scale the red parallelogram to the green one, given the blue triangle so that both top corners of the green parallelogram touches the triangle?

Answer 1

Draw a line, parallel to the sides of the parallelogram, through the apex of the triangle. The point where this line intersects the base of the triangle is the centre of the scaling operation, so if you draw lines from there to the corners of the red parallelogram they will go through the corners of the green parallelogram as well exactly where they intersect the sides of the triangle.

Answer 2

Start with the red parallelogram, draw on its upper side a yellow triangle similar to blue triangle, i.e., with parallel sides. Extend the sides until they reach the base of the parallelogram extended. Now compare the yellow triangle with the blue triangle. Keeping the same ratio, draw a smaller green parallelogram by first dividing the sides of the blue triangle in the same ratio, as in Euclid Book 6 Prop. 10.

Answer 3

HINT:

The idea: on the base on the triangle, build the parallelogram $below$ the triangle. Now, the parallelogram inside the triangle has to be homothetic to this one. Therefore, join the opposite vertex of the triangle with the other two of the vertices of the parallelogram and look at the intersections with the base.

$\bf{Added:}$ Similarly you can scale a prism to fit inside a pyramid.