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rectangle wanted, 4 circles inside a circle given

Source: Dutch NMO 2003 p4

September 21, 2019
geometryrectanglecircles

Problem Statement

In a circle with center MM, two chords ACAC and BDBD intersect perpendicularly. The circle of diameter AMAM intersects the circle of diameter BMBM besides MM also in point PP. The circle of diameter BMBM intersects the circle with diameter CMCM besides MM also in point QQ. The circle of diameter CMCM intersects the circle of diameter DMDM besides MM also in point RR. The circle of diameter DMDM intersects the circle of diameter AMAM besides MM also in point SS. Prove that quadrilateral PQRSPQRS is a rectangle.
[asy] unitsize (3 cm);
pair A, B, C, D, M, P, Q, R, S;
M = (0,0); A = dir(170); C = dir(10); B = dir(120); D = dir(240);
draw(Circle(M,1)); draw(A--C); draw(B--D); draw(Circle(A/2,1/2)); draw(Circle(B/2,1/2)); draw(Circle(C/2,1/2)); draw(Circle(D/2,1/2)); P = (A + B)/2; Q = (B + C)/2; R = (C + D)/2; S = (D + A)/2;
dot("AA", A, A); dot("BB", B, B); dot("CC", C, C); dot("DD", D, D); dot("MM", M, E); dot("PP", P, SE); dot("QQ", Q, SE); dot("RR", R, NE); dot("SS", S, NE); [/asy]
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