MathDB
A Circle and a Product

Source:

June 8, 2009
geometryratiopower of a point

Problem Statement

Circle O O has diameters AB AB and CD CD perpendicular to each other. AM AM is any chord intersecting CD CD at P P. Then APAM AP\cdot AM is equal to: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm); pair O = origin; pair A = (-1,0); pair B = (1,0); pair C = (0,1); pair D = (0,-1); pair M = dir(45); pair P = intersectionpoint(O--C,A--M);
draw(Circle(O,1)); draw(A--B); draw(C--D); draw(A--M);
label("AA",A,W); label("BB",B,E); label("CC",C,N); label("DD",D,S); label("MM",M,NE); label("OO",O,NE); label("PP",P,NW);[/asy]<spanclass=latexbold>(A)</span> AOOB<spanclass=latexbold>(B)</span> AOAB<spanclass=latexbold>(C)</span> CPCD<spanclass=latexbold>(D)</span> CPPD <span class='latex-bold'>(A)</span>\ AO\cdot OB \qquad <span class='latex-bold'>(B)</span>\ AO\cdot AB\qquad <span class='latex-bold'>(C)</span>\ CP\cdot CD \qquad <span class='latex-bold'>(D)</span>\ CP\cdot PD\qquad <spanclass=latexbold>(E)</span> COOP <span class='latex-bold'>(E)</span>\ CO\cdot OP