maths 趕！10點！

This is known as Van Aubel's Theorem:
Sorry I don't know how to post photo. Please draw yourself.)

Proof:
Let X be the midpoint of BD.
Let Y be the intersection of RP and SQ

1) [Show: XS = XP and ∠SXP=90]
Let E = upper left corner of the P-square, Let F = upper left corner of the S-square.
Then △EAD≡△BAF
(SAS: EA=AB, FA=FD,∠BAF=∠FAE+90=∠EAD)

So FB=ED
Let G=ED intersect FA. Let H = FB intersect ED
Then ∠HFG = ∠EDA, ∠FGH=∠DGA
so ∠FHD=∠FAD=90
Hence FB⊥ ED.

Finally by midpoint theorem, SX = 1/2 FB = 1/2 ED = XP, and SX // FB ⊥ ED // XP
so ∠SXP =90

2) Similarly, QX=XR, and ,∠QXR=90

3) So △PXR ≡△SXQ
(SAS: QX=XR, SX=XP, ∠PXR = ∠PXQ+90=∠SXQ)

4) So RP = SQ (Done part 1)

5) ∠SQX = ∠PRX
Let Z=PR intersect XQ
Then ∠RZX = ∠YZQ

So ∠ZYQ=∠ZXR = 90

=> RP ⊥SQ