In his book The Science of Education, Gattegno describes various ways of knowing.
Try solving the problem below, paying particular attention to the moments that you know you are getting closer to the answer, especially the precise moment that you know you have the solution.
How exactly do you know, at these moments, that you know? What are the feelings associated with knowing? Can you sense a mobilisation of energy in these moments?
P is the centre of the square constructed on the hypotenuse AC of the right-angled triangle ABC.
Prove that BP bisects angle ABC.