Square From Nowhere: What is this about?
A Mathematical Droodle
What if applet does not run? 
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander BogomolnyThe applet suggests the following construction of a square:
Let points P and Q lie on a segment AR and satisfy AP = QR. At points P, Q, R erect perpendiculars PD, QB, RC to AR such that PD = PR, QB = QR, and RC = PQ. Then the quadrilateral ABCD is a square. 
What if applet does not run? 
The proof below assumes that P and Q lie between A and R. This need not necessarily be the case, but if P and Q are without AR, the proof has to be slightly adjusted.
First consider right triangles APD and AQB:
AP = QR = QB, PD = PR = AQ. 
It follows that ΔAPD = ΔAQB. In particular,
Further, define T on the extension of RC such that BTAR and consider ΔBTC.
BT = QR = AP, 
and

Hence ΔBTC equals the other two. In particular,
References
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny70185136