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

 

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Explanation

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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:

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: PQ2 = PR·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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What if applet does not run?

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 PR/PQ = PQ/PS.

A different solution was found by Vo Duc Dien.

References

  1. T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, pp. 6-7.

Related material
Read more...

  • The Means
  • Averages, Arithmetic and Harmonic Means
  • Arithmetic and Geometric Means
  • Geometric Meaning of the Geometric Mean
  • Short Construction of the Geometric Mean
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