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Tangents, Perpendiculars and Geometric Mean: What Is This About?
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The 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:
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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.
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We conclude that, in ARPQ,
 RAP = RQP and alsoPAQ = PRQ
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as inscribed and subtended by the same chord. Similarly, in BSPQ,
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SBP = SQP and also
PBQ = PSQ.
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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
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RQP = RAP = ABP = PBQ = PSQ,
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so that
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RQP = PSQ.
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Similarly,
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SQP = PRQ.
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Thus triangles PRQ and PQS are similar and therefore
References
- T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, pp. 6-7.
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