MathDB

2

Part of 2009 ITAMO

Problems(2)

ABCD is a square with centre O

Source: itamo 2009 p2

2/17/2012
ABCDABCD is a square with centre OO. Two congruent isosceles triangle BCJBCJ and CDKCDK with base BCBC and CDCD respectively are constructed outside the square. let MM be the midpoint of CJCJ. Show that OMOM and BKBK are perpendicular to each other.
geometrygeometric transformationrotationgeometry proposed
T,R,K are collinear

Source: ItaMO 2009, p5

2/23/2012
Let ABCABC be an acute-angled scalene triangle and Γ\Gamma be its circumcircle. KK is the foot of the internal bisector of ∠BAC\angle BAC on BCBC. Let MM be the midpoint of the arc BCBC containing AA. MKMK intersect Γ\Gamma again at A′A'. TT is the intersection of the tangents at AA and A′A'. RR is the intersection of the perpendicular to AKAK at AA and perpendicular to A′KA'K at A′A'. Show that T,RT, R and KK are collinear.
geometrycircumcircleprojective geometrygeometry proposed