Hiroshi Haruki's Lemma

Hiroshi Haruki's Lemma: What is this about?
A Mathematical Droodle

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The applet may suggest the following assertion:

Assume AB and CD are two chords and P a point on a circle. Let Q and R be the intersections of PC and PD with AB. Draw circumcircles of triangles PRC and PQD and let AB intersect the two circles in F and E, respectively. Then

(1)	BE = AF, and their length does not depend on the position of P on the given circle.

The latter is an unintended consequence of the proof of Hiroshi Haruki's Lemma [Honsberger]:

(2)	For AB, CD, P, Q, R as above, AQ·RB / QR does not depend on the position of P on the given circle.

Proof

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In the circumcircle PQD, if P lies in the major arc CD, the angles QPD and QED are equal as subtending the same chord QD. As P traverses the major arc CD , the angle at E (AED) remains the same, which means that the point E remains fixed. (This is in fact true even if P lies on the minor arc CD.) So that BE = const, independent of the position of P on the circle.

We apply the intersecting Chords Theorem to PD crossed by AE in two circles:

PR·RD = QR·RE and PR·RD = AR·RB,

from which

QR·(RB + BE) = (AQ + QR)·RB,

or QR·BE = AQ·RB, and finally

(3)	AQ·RB / QR = BE.

This proves (2) and also (1), since a similar argument in the circle PRC shows that F is independent of P and, in addition, AF = AQ·RB / QR, the same expression as in (3).

The lemma leads to elegant proofs of the Butterfly and Two Butterflies theorems.

References

R. Honsberger, The Butterfly Problem and Other Delicacies from the Noble Art of Euclidean Geometry I, TYCMJ, 14 (1983), pp. 2-7.
R. Honsberger, Mathematical Diamonds, MAA, 2003, pp. 136-140

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