How can water exist in three states in freezing point? Freezing point is temperature on which vapor pressures of solid and liquid are equal. This is definition from my book. How can water ice and vapor of those co-exist when water that is only possible in triple point? How can vapor of those even  exist when in freezing point (melting point) there is equilibrium between solid and liquid? Is it when phases co-exist there is always  equilibrium between them question is only is it placed more towards one phase or the other?
If you only have a single component, water, present, then the liquid, vapor, and solid can only exist at the triple point.  If there is a second component present, say air, the phase rule tells us that there is an additional degree of freedom, pressure.  So, at a total pressure of 1 atm, the freezing point is 0 C (slightly lower than the triple point temperature), and you have liquid water, solid water, and, in the gas phase, water vapor mixed with air.