Kinetic Temperaturea and molecular velocities

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If I have ice, water and vapour at the triple point and they are in equilibrium at a temperature of 273.16. Does this mean that the average kinetic energy of the particles is the same in each state? If so how can the molecules in the water be moving with the same mean squared speed as the molecules in the vapour? Thanks for any help
 
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Thanks for responding Matterwave

LatentHeat1_zps76e36554.jpg


Where this question has come from is a discussion I was having about temperature increase as water changes from the solid to the gaseous state. As the water changes from D to E, If temperature is defined as average KE =3/2kT does this imply that the KE and hence the translational mean squared speeds of the liquid at D and the gas at E are the same if they are at the same temperature?
 
Does kinetic temperature real because of molecular moving?
Hint: something else moving?