Re: [EMHL] a quadrilateral problem
- Dear Mustafa
>ABCD is a convex quadrilateral with area 18.is with other words the 1st problem of the 18 IMO (1976).
>If |AB|+|BD|+|DC|=12 then |AC|=?
>This problem was asked in a math olympiad elimination in
> Turkey.I couldn't solve it. Could you also try?
>Are these given informations enough to solve?
If a = AB, b = BD, c = DC, a+b+c=12
x = angABD, y = angBDC then
36 = 2(ABCD)
<= b(a+c) <= ((a+b+c)/2)² = 36
So we must have only equality i.e.
sin(x) = sin(y) = 1, b = a+c = 6.
Hence x = y = 90 and if the parallel from A to BD
meets CD at E the right angled triangle AEC
has AE = b = 6, EC = a+c = 6 and AC = 6sqrt(2).