MathDB

2

Part of 2009 Italy TST

Problems(2)

the locus of $P$

Source: Italy TST 2009 p2

3/10/2012
ABCABC is a triangle in the plane. Find the locus of point PP for which PA,PB,PCPA,PB,PC form a triangle whose area is equal to one third of the area of triangle ABCABC.
geometryparameterizationgeometry unsolved
two triangles have equal circumradii

Source: Italy TST 2009 p5

3/10/2012
Two circles O1O_1 and O2O_2 intersect at M,NM,N. The common tangent line nearer to MM of the two circles touches O1,O2O_1,O_2 at A,BA,B respectively. Let C,DC,D be the symmetric points of A,BA,B with respect to MM respectively. The circumcircle of triangle DCMDCM intersects circles O1O_1 and O2O_2 at points E,FE,F respectively which are distinct from MM. Prove that the circumradii of the triangles MEFMEF and NEFNEF are equal.
geometrycircumcircletrapezoidgeometric transformationreflectionpower of a pointradical axis