Help on understanding proof

[junqian2020]

Hi.. I would like some help regarding this problem...
Given Triangle DEF. Let points A and B, drawn externally to Triangle DEF, be the center of squares having DE and EF as sides, respectively. Let C be the midpoint of DF. Show that performing R_B,90^O(R_A,90^O(X)) is the same as performing R_C,180^O(X), with X being an arbitrary point in the plane.

I observed that performing R_B,90^O(R_A,90^O(D)) and R_C,180^O(D) produced the same image for D. I am thinking of using the concept that a rotation at some center A with an angle X and followed by another rotation about a point B with an an angle Y, is a rotation at some point C with an angle X + Y (as long as X + Y is not a multiple of 2pi). However, it seems that just saying, "Since R_B,90^O(R_A,90^O(X)) is a composite rotation with a total angle of 180^O (not a multiple of 2pi) then there exists a point C such that R_C,180^O(X) produces the same rotation." is not enough since I did not use the fact that D,E and F forms a triangle and that A and B are centers of two different squares. I would like to ask for help regarding this. Thank you.

 

[Cubist]

However, it seems that just saying, "Since R_B,90^O(R_A,90^O(X)) is a composite rotation with a total angle of 180^O (not a multiple of 2pi) then there exists a point C such that R_C,180^O(X) produces the same rotation." is not enough since I did not use the fact that D,E and F forms a triangle and that A and B are centers of two different squares.

I agree, this is not enough. And your statement uses the pre-existing label C but you shouldn't assume that C is the centre of rotation. Use a label like G in such a statement, and then prove that C and G are at the same coordinates.

Here's a diagram...



FYI: The question seems valid. I copied the shape in red and blue colours in my graphics program. I put the two shapes directly on top of each other. The blue one was then rotated 90° clockwise around point A. Then 90° clockwise around B. Not definitive proof, but look what happened...



Have you got any ideas about how to start? It might be sensible to assign some Cartesian coordinates to the points. Why not choose C being at the origin? Then the coordinates of F and D will be (a,b) and (?,?). Continue, and please post back with your work...