Square From Nowhere

Square From Nowhere: What is this about?
A Mathematical Droodle

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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:

AP = QR = QB,
PD = PR = AQ.

It follows that ΔAPD = ΔAQB. In particular, AD = AB and ∠ADP = ∠BAQ. Therefore, ∠BAD is right.

Further, define T on the extension of RC such that BT||AR and consider ΔBTC.

BT = QR = AP,

and

CT	= RC + RT
	= PQ + QB
	= PQ + QR
	= PR
	= PD.

Hence ΔBTC equals the other two. In particular, BC = AB and ∠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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