MathDB

2

Part of 1999 IberoAmerican

Problems(2)

14th ibmo - cuba 1999/q2.

Source: Spanish Communities

4/16/2006
Given two circle MM and NN, we say that MM bisects NN if they intersect in two points and the common chord is a diameter of NN. Consider two fixed non-concentric circles C1C_1 and C2C_2. a) Show that there exists infinitely many circles BB such that BB bisects both C1C_1 and C2C_2. b) Find the locus of the centres of such circles BB.
geometry unsolvedgeometry
14th ibmo - cuba 1999/q5.

Source: Spanish Communities

4/16/2006
An acute triangle △ABC\triangle{ABC} is inscribed in a circle with centre OO. The altitudes of the triangle are AD,BEAD,BE and CFCF. The line EFEF cut the circumference on PP and QQ. a) Show that OAOA is perpendicular to PQPQ. b) If MM is the midpoint of BCBC, show that AP2=2AD⋅OMAP^2=2AD\cdot{OM}.
geometrycircumcirclesymmetrygeometry unsolved