MathDB

4

Part of 1992 Irish Math Olympiad

Problems(2)

Diameter of Circumcircle

Source: 1992 IrMO Paper 1 Problem 4

10/2/2017
In a triangle ABCABC, the points A’A’, B’B’ and C’C’ on the sides opposite AA ,BB and CC, respectively, are such that the lines AA’AA’, BB’BB’ and CC’CC’ are concurrent. Prove that the diameter of the circumscribed circle of the triangle ABCABC equals the product ∣AB’∣⋅∣BC’∣⋅∣CA’∣|AB’|\cdot |BC’|\cdot |CA’| divided by the area of the triangle A’B’C’A’B’C’.
geometrycircumcircle
Area of Pentagon

Source: 1992 IrMO Paper 2 Problem 4

10/2/2017
A convex pentagon has the property that each of its diagonals cuts off a triangle of unit area. Find the area of the pentagon.
geometry