# 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

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

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

### 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