MathDB

3

Part of 2007 Irish Math Olympiad

Problems(2)

fixed point

Source: Ireland 2007

7/6/2009
The point P P is a fixed point on a circle and Q Q is a fixed point on a line. The point R R is a variable point on the circle such that P,Q, P,Q, and R R are not collinear. The circle through P,Q, P,Q, and R R meets the line again at V V. Show that the line VR VR passes through a fixed point.
geometrycircumcirclegeometric transformationreflectiongeometry proposed
identity

Source: Ireland 2007

7/6/2009
Let ABC ABC be a triangle the lengths of whose sides BC,CA,AB, BC,CA,AB, respectively, are denoted by a,b, a,b, and c c. Let the internal bisectors of the angles ∠BAC,∠ABC,∠BCA, \angle BAC, \angle ABC, \angle BCA, respectively, meet the sides BC,CA, BC,CA, and AB AB at D,E, D,E, and F F. Denote the lengths of the line segments AD,BE,CF AD,BE,CF by d,e, d,e, and f f, respectively. Prove that: def\equal{}\frac{4abc(a\plus{}b\plus{}c) \Delta}{(a\plus{}b)(b\plus{}c)(c\plus{}a)} where Δ \Delta stands for the area of the triangle ABC ABC.
geometryinequalitiesgeometry proposed