Pentagon in a Semicircle

This is Problem 1 from the 2010 USAMO

Let AXYZB be a convex pentagon inscribed in a semicircle of diameter AB. Denote by P, Q, R, S the feet of the perpendiculars from Y onto lines AX, BX, AZ, BZ, respectively. Prove that the acute angle formed by lines PQ and RS is half the size of ∠XOZ, where O is the midpoint of segment AB.


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Solution

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Let AXYZB be a convex pentagon inscribed in a semicircle of diameter AB. Denote by P, Q, R, S the feet of the perpendiculars from Y onto lines AX, BX, AZ, BZ, respectively. Prove that the acute angle formed by lines PQ and RS is half the size of ∠XOZ, where O is the midpoint of segment AB.


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Solution 1

(A write-up in the Mathematics Magazine.)

Let I be the foot of the perpendicular from Y to line AB. We note the P, Q, I are the feet of the perpendiculars from Y to the sides of triangle ABX. Because Y lies on the circumcircle of triangle ABX, points P, Q, I are collinear, by Simsonfs theorem. Likewise, points S, R, I are collinear. We need to show that ∠XOZ = 2∠PIS or

∠PIS= ½∠XOZ = ½(arc XZ) = ½(arc XY) + ½(arc YZ)
 = ∠XAY + ∠ZBY
 = ∠PAY + ∠SBY.

Because ∠PIS = ∠PIY + ∠SIY, it suffices to prove that

∠PIY = ∠PAY and ∠SIY = ∠SBY;

that is, to show that quadrilaterals APYI and BSYI are cyclic, which is evident, because ∠APY = ∠AIY = 90° and ∠BIY = ∠BSY = 90°.

Solution 2

(By Vo Duc Dien)

Let AZ intercept BX at C, PQ and RS intercept at I. The acute angle formed by lines PQ and RS is

∠PIS = ∠PQY + ∠SRY - ∠QYR = ∠PQY + ∠SRY - ∠RCB

because angles QYR and RCB have pairwise perpendicular sides.

But ∠RCB subtends arcs AX and BZ; ∠PQY = ∠PXY subtends arc AY; ∠SRY = ∠SZY subtends arc BY.

Therefore, ∠PIS subtends the arc(AY) + arc(BY) - arc(AX) - arc(BZ) = arc(XZ) = ½∠XOZ.

References

  1. JACEK FABRYKOWSKI, STEVEN R. DUNBAR, 39th USA Mathematical Olympiad - 1st USA Junior Mathematical Olympiad, Math. Mag. 83 (2010) 313-319

Related material
Read more...

Simson Line - the simson

  • Simson Line: Introduction
  • Simson Line
  • Three Concurrent Circles
  • 9-point Circle as a locus of concurrency
  • Miquel's Point
  • Circumcircle of Three Parabola Tangents
  • Angle Bisector in Parallelogram
  • Simsons and 9-Point Circles in Cyclic Quadrilateral
  • Reflections of a Point on the Circumcircle
  • Simsons of Diametrically Opposite Points
  • Simson Line From Isogonal Perspective
  • Simson Line in Disguise
  • Two Simsons in a Triangle
  • Carnot's Theorem
  • A Generalization of Simson Line

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