MathDB
Equilateral Triangle in a Circle

Source:

June 8, 2009

Problem Statement

Let ABC ABC be an equilateral triangle inscribed in circle O O. M M is a point on arc BC BC. Lines AM \overline{AM}, BM \overline{BM}, and CM \overline{CM} are drawn. Then AM AM is: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm);
pair O = origin; pair B = (1,0); pair C = dir(120); pair A = dir(240); pair M = dir(90 - 18);
draw(Circle(O,1)); draw(A--C--M--B--cycle); draw(B--C); draw(A--M); dot(O);
label("AA",A,SW); label("BB",B,E); label("MM",M,NE); label("CC",C,NW); label("OO",O,SE);[/asy]<spanclass=latexbold>(A)</span> equal to BM+CM<spanclass=latexbold>(B)</span> less than BM+CM<spanclass=latexbold>(C)</span> greater than BM+CM <span class='latex-bold'>(A)</span>\ \text{equal to }{BM + CM}\qquad <span class='latex-bold'>(B)</span>\ \text{less than }{BM + CM}\qquad <span class='latex-bold'>(C)</span>\ \text{greater than }{BM + CM}\qquad <spanclass=latexbold>(D)</span> equal, less than, or greater than BM+CM, depending upon the position of M <span class='latex-bold'>(D)</span>\ \text{equal, less than, or greater than }{BM + CM}\text{, depending upon the position of }{ {M}\qquad} <spanclass=latexbold>(E)</span> none of these <span class='latex-bold'>(E)</span>\ \text{none of these}