MathDB

2

Part of 2001 Iran MO (2nd round)

Problems(2)

B'M perp to A'C' - Iran NMO 2001 (Second Round) - Problem2

Source:

10/4/2010
Let ABCABC be an acute triangle. We draw 33 triangles B′AC,C′AB,A′BCB'AC,C'AB,A'BC on the sides of ΔABC\Delta ABC at the out sides such that: ∠B′AC=∠C′BA=∠A′BC=30∘   ,   ∠B′CA=∠C′AB=∠A′CB=60∘ \angle{B'AC}=\angle{C'BA}=\angle{A'BC}=30^{\circ} \ \ \ , \ \ \ \angle{B'CA}=\angle{C'AB}=\angle{A'CB}=60^{\circ} If MM is the midpoint of side BCBC, prove that B′MB'M is perpendicular to A′C′A'C'.
geometry proposedgeometry
How much is angleA - Iran NMO 2001 (Second Round) - Problem5

Source:

10/4/2010
In triangle ABCABC, AB>ACAB>AC. The bisectors of ∠B,∠C\angle{B},\angle{C} intersect the sides AC,ABAC,AB at P,QP,Q, respectively. Let II be the incenter of ΔABC\Delta ABC. Suppose that IP=IQIP=IQ. How much isthe value of ∠A\angle A?
geometryincentertrigonometrycircumcircleperpendicular bisectorgeometry proposed