MathDB

2

Part of 2005 Iran Team Selection Test

Problems(2)

PT is perpendicular to XY

Source: Iran TST 2005

4/20/2005
Assume ABCABC is an isosceles triangle that AB=ACAB=AC Suppose PP is a point on extension of side BCBC. XX and YY are points on ABAB and ACAC that: PX∣∣AC , PY∣∣ABPX || AC \ , \ PY ||AB Also TT is midpoint of arc BCBC. Prove that PT⊥XYPT \perp XY
geometrycircumcircleratioparallelogramsymmetrygeometric transformationreflection
O is constant

Source: Iran TST 2005

4/20/2005
Suppose there are nn distinct points on plane. There is circle with radius rr and center OO on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move OO to the baricenter of the point in the circle. Prove that location of OO is constant after some steps.
functiongeometry proposedgeometry