MathDB

2

Part of 1997 IberoAmerican

Problems(2)

12nd ibmo - mexico 1997/q2.

Source: Spanish Communities

4/22/2006
In a triangle ABCABC, it is drawn a circumference with center in the incenter II and that meet twice each of the sides of the triangle: the segment BCBC on DD and PP (where DD is nearer two BB); the segment CACA on EE and QQ (where EE is nearer to CC); and the segment ABAB on FF and RR ( where FF is nearer to AA).
Let SS be the point of intersection of the diagonals of the quadrilateral EQFREQFR. Let TT be the point of intersection of the diagonals of the quadrilateral FRDPFRDP. Let UU be the point of intersection of the diagonals of the quadrilateral DPEQDPEQ.
Show that the circumcircle to the triangle △FRT\triangle{FRT}, △DPU\triangle{DPU} and △EQS\triangle{EQS} have a unique point in common.
geometryincentercircumcircleinradiustrapezoidgeometric transformationrotation
12nd ibmo - mexico 1997/q5.

Source: Spanish Communities

4/22/2006
In an acute triangle △ABC\triangle{ABC}, let AEAE and BFBF be highs of it, and HH its orthocenter. The symmetric line of AEAE with respect to the angle bisector of ∢A\sphericalangle{A} and the symmetric line of BFBF with respect to the angle bisector of ∢B\sphericalangle{B} intersect each other on the point OO. The lines AEAE and AOAO intersect again the circuncircle to △ABC\triangle{ABC} on the points MM and NN respectively.
Let PP be the intersection of BCBC with HNHN; RR the intersection of BCBC with OMOM; and SS the intersection of HRHR with OPOP. Show that AHSOAHSO is a paralelogram.
geometrycircumcircletrapezoidparallelogramvectortrigonometryangle bisector