MathDB

2

Part of 1998 Iran MO (2nd round)

Problems(2)

Value of angle BAC - Iran NMO 1998 (Second Round) Problem2

Source:

10/4/2010
Let ABCABC be a triangle. II is the incenter of ΔABC\Delta ABC and DD is the meet point of AIAI and the circumcircle of ΔABC\Delta ABC. Let E,FE,F be on BD,CDBD,CD, respectively such that IE,IFIE,IF are perpendicular to BD,CDBD,CD, respectively. If IE+IF=AD2IE+IF=\frac{AD}{2}, find the value of ∠BAC\angle BAC.
geometryincentercircumcircletrigonometrygeometry proposed
AQ+CQ=BP - Iran NMO 1998 (Second Round) Problem5

Source:

10/4/2010
Let ABCABC be a triangle and AB<AC<BCAB<AC<BC. Let D,ED,E be points on the side BCBC and the line ABAB, respectively (AA is between B,EB,E) such that BD=BE=ACBD=BE=AC. The circumcircle of ΔBED\Delta BED meets the side ACAC at PP and BPBP meets the circumcircle of ΔABC\Delta ABC at QQ. Prove that: AQ+CQ=BP. AQ+CQ=BP.
geometrycircumcircleperpendicular bisectorcyclic quadrilateralgeometry proposed