MathDB

8

Part of 1997 Taiwan National Olympiad

Problems(1)

$OD.OE.OF\geq 8R^3$

Source: 6-th Taiwanese Mathematical Olympiad 1997

1/18/2007
Let OO be the circumcenter and RR be the circumradius of an acute triangle ABCABC. Let AOAO meet the circumcircle of OBCOBC again at DD, BOBO meet the circumcircle of OCAOCA again at EE, and COCO meet the circumcircle of OABOAB again at FF. Show that OD.OE.OF≥8R3OD.OE.OF\geq 8R^{3}.
geometrycircumcircletrigonometrygeometric transformationreflectionincenterinequalities