Delivered energy of a RLC circuit - proof
By simple observation, when the charged capacitor connects to the right hand circuit, all the stored energy will be delivered to the resistors over time. It has nowhere else to go. That stored energy will be dissipated as heat.

Given that some proportion of that energy will be delivered to both resistors, is it clear to see that if they were equal in value, each would receive 50% of the original stored energy. So, do you really need any more help understanding the resistance ratio in your 2nd equation?