MathDB

3

Part of 1992 IberoAmerican

Problems(2)

An equilateral triangle

Source: Iberoamerican Olympiad 1992, Problem 3

5/14/2007
Let ABCABC be an equilateral triangle of sidelength 2 and let ω\omega be its incircle. a) Show that for every point PP on ω\omega the sum of the squares of its distances to AA, BB, CC is 5. b) Show that for every point PP on ω\omega it is possible to construct a triangle of sidelengths APAP, BPBP, CPCP. Also, the area of such triangle is 34\frac{\sqrt{3}}{4}.
geometrycircumcircletrigonometryinradiusinequalitiesincentergeometry proposed
Nice geometric inequality

Source: Iberoamerican Olympiad 1992, Problem 6

5/14/2007
In a triangle ABCABC, points A1A_{1} and A2A_{2} are chosen in the prolongations beyond AA of segments ABAB and ACAC, such that AA1=AA2=BCAA_{1}=AA_{2}=BC. Define analogously points B1B_{1}, B2B_{2}, C1C_{1}, C2C_{2}. If [ABC][ABC] denotes the area of triangle ABCABC, show that [A1A2B1B2C1C2]≥13[ABC][A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}] \geq 13 [ABC].
inequalitiesgeometrytrigonometrycircumcircleinequalities proposed