MathDB

3

Part of 2007 Germany Team Selection Test

Problems(3)

Prove that PA*BC = PB*AC = PC*AB

Source: VAIMO 2007, P3

1/3/2009
A point P P in the interior of triangle ABC ABC satisfies \angle BPC \minus{} \angle BAC \equal{} \angle CPA \minus{} \angle CBA \equal{} \angle APB \minus{} \angle ACB. Prove that \bar{PA} \cdot \bar{BC} \equal{} \bar{PB} \cdot \bar{AC} \equal{} \bar{PC} \cdot \bar{AB}.
geometry unsolvedgeometry
Sine inequality greater than twice triangle area

Source: AIMO 2007, TST 5, P3

1/11/2009
Let ABC ABC be a triangle and P P an arbitrary point in the plane. Let α,β,γ \alpha, \beta, \gamma be interior angles of the triangle and its area is denoted by F. F. Prove: \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F When does equality occur?
trigonometryinequalitiesgeometrysearchgeometry unsolved
Ratio of circumcircle radius to incircle diameter

Source: AIMO 2007, TST 6, P3

1/11/2009
In triangle ABC ABC we have a≥b a \geq b and a≥c. a \geq c. Prove that the ratio of circumcircle radius to incircle diameter is at least as big as the length of the centroidal axis sa s_a to the altitude aa. a_a. When do we have equality?
ratiogeometrycircumcirclegeometry unsolved