MathDB

5

Part of 2006 Iran Team Selection Test

Problems(2)

Question 5

Source: Iran TST 2006

4/18/2006
Let ABCABC be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to AA. Suppose that the outer inscribed circle with respect to AA touches BC,AC,ABBC,AC,AB at M,N,LM,N,L. Prove that OO (Center of circumcircle) is the orthocenter of MNLMNL.
geometrycircumcircletrigonometryparallelogramangle bisectorgeometry proposed
Question 5

Source: Iran TST 2006

4/30/2006
Let ABCABC be an acute angle triangle. Suppose that D,E,FD,E,F are the feet of perpendicluar lines from A,B,CA,B,C to BC,CA,ABBC,CA,AB. Let P,Q,RP,Q,R be the feet of perpendicular lines from A,B,CA,B,C to EF,FD,DEEF,FD,DE. Prove that 2(PQ+QR+RP)≥DE+EF+FD 2(PQ+QR+RP)\geq DE+EF+FD
geometrycircumcircletrigonometryinequalitiesfunctionincentergeometry proposed