MathDB

Cyclic quadrilateral with cosecant equality

Source: IMO LongList 1988, Spain 2, Problem 71 of ILL

November 9, 2005

trigonometrygeometry unsolvedgeometry

Problem Statement

The quadrilateral

A_1A_2A_3A_4

is cyclic, and its sides are

a_1 = A_1A_2, a_2 = A_2A_3, a_3 = A_3A_4

and

a_4 = A_4A_1.

The respective circles with centres

I_i

and radii

r_i

are tangent externally to each side

a_i

and to the sides

a_{i+1}

and

a_{i-1}

extended. (

a_0 = a_4

). Show that

\prod^4_{i=1} \frac{a_i}{r_i} = 4 \cdot (\csc (A_1) + \csc (A_2) )^2.

Back to Problems View on AoPS