MathDB

5

Part of 1991 Dutch Mathematical Olympiad

Problems(1)

triangle

Source: Netherlands 1991

6/28/2009
Let H H be the orthocenter, O O the circumcenter, and R R the circumradius of an acute-angled triangle ABC ABC. Consider the circles ka,kb,kc,kh,k k_a,k_b,k_c,k_h,k, all with radius R R, centered at A,B,C,H,M, A,B,C,H,M, respectively. Circles ka k_a and kb k_b meet at M M and F F; ka k_a and kc k_c meet at M M and E E; and kb k_b and kc k_c meet at M M and D D. (a) (a) Prove that the points D,E,F D,E,F lie on the circle kh k_h. (b) (b) Prove that the set of the points inside kh k_h that are inside exactly one of the circles ka,kb,kc k_a,k_b,k_c has the area twice the area of △ABC \triangle ABC.
geometrycircumcircleparallelogramgeometry proposed