Tangents, Perpendiculars and Geometric Mean: What Is This About?
A Mathematical Droodle

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Copyright © 19962018 Alexander BogomolnyThe applet illustrates a problem that exploits relationships between angles and arcs in a circle. Hidden below the surface, there is a segment that serves as the geometric mean of other two:
Let AB be a chord in a circle and P a point on the circle. Let Q be the feet of the perpendicular from P to AB, and R and S the feet of the perpendiculars from P to the tangents to the circle at A and B. Prove that PQ is the geometric mean of PR and PS:
The problem does not require additional construction beyond the observation that quadrilaterals ARPQ and BSPQ are cyclic. This is because each contains a pair of right angles opposite each other.

We conclude that, in ARPQ,
∠RAP = ∠RQP and also ∠PAQ = ∠PRQ 
as inscribed and subtended by the same chord. Similarly, in BSPQ,
∠SBP = ∠SQP and also ∠PBQ = ∠PSQ. 
But in the given circle the inscribed ∠ABP is subtended by chord AP that forms with the tangent at A an equal ∠RAP. It follows that
∠RQP = ∠RAP = ∠ABP = ∠PBQ = ∠PSQ, 
so that
∠RQP = ∠PSQ. 
Similarly,
∠SQP = ∠PRQ. 
Thus triangles PRQ and PQS are similar and therefore
A different solution was found by Vo Duc Dien.
References
 T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, pp. 67.
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Copyright © 19962018 Alexander Bogomolny69257822