Cross's theorem
First try and explain why (prove) it is true yourself. But if you get stuck, have a look at one of my student's (unedited) proofs at Gin Chou's proof
Similar rectangles on sides of quadrilateral
Similar parallelograms on sides of quadrilateral-1
Similar parallelograms on sides of quadrilateral-2
Similar cyclic quadrilaterals on sides of quadrilateral
Continuing constructing squares on the sides