MathDB

4

Part of 1996 Irish Math Olympiad

Problems(2)

easy geometry

Source: Ireland 1996

7/1/2009
Let F F be the midpoint of the side BC BC of a triangle ABC ABC. Isosceles right-angled triangles ABD ABD and ACE ACE are constructed externally on AB AB and AC AC with the right angles at D D and E E. Prove that the triangle DEF DEF is right-angled and isosceles.
geometrygeometry unsolved
concurrent lines

Source: Ireland 1996

7/1/2009
In an acute-angled triangle ABC ABC, D,E,F D,E,F are the feet of the altitudes from A,B,C A,B,C, respectively, and P,Q,R P,Q,R are the feet of the perpendiculars from A,B,C A,B,C onto EF,FD,DE EF,FD,DE, respectively. Prove that the lines AP,BQ,CR AP,BQ,CR are concurrent.
geometrycircumcirclegeometry unsolved