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Square From Nowhere: What is this about?
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The 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. |
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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:
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AP = QR = QB, PD = PR = AQ. |
It follows that 
APD = AQB. In particular, 
ADP = BAQ. Therefore, BAD is right.
Further, define T on the extension of RC such that BT||AR and consider BTC.
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BT = QR = AP, |
and
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Hence BTC equals the other two. In particular, ABC is also right. This does show that ABCD is a square.
References
- N. A. Court, Mathematics in Fun and in Earnest, Mentor Books, 1961
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