Prove In Plato's Line of Knowledge that Belief part of whole is equal to Learned part of the whole

Question

Many people have many interpretations of Plato's Line of Knowledge. This question is/

asks about the math/algebra to prove the middle pieces equal.

the whole = a person's body of knowledge on any particular subject that has been studied

common sense = what hasn't been formally studied=A

belief = what must almost certainly be true based on person's firm beliefs=B

common sense+belief=daily input ,,,A+B=C

learned = what is studied=D

mastered = the articulation of knowledge=E

learned+mastered = schooling,,,,D+E=F

the whole = (common sense)+belief+learned+mastered=A+B+D+E=C+F

assume the following ratio for each division indicated are equal and not equal to 50% (which makes the problem trivial).

[belief/(common sense)]=[mastered/learned]=[schooling/(daily input)],,,B/A=E/D=F/C

prove no matter what the ratio is,

belief = learned,,,,B=D

in terms of size of the part of the whole

Answer 1

Given our three equalities of B/A = E/D = F/C, we can rearrange variables to get something that's easier to work with.

From B/A = F/C, we get BC = FA by cross multiplying.

Note that F = D+E, so by substituting we see that BC = FA = (D+E)A = DA + AE.

So BC = DA + AE and thus rearranging we get AE = BC - DA.

Note that from the equality B/A = E/D, we get that BD = AE.

So now we get BD = BC - DA

Note that C = A+B, so substituting gives us BD = B(A+B) - DA.

Add DA to both sides and we have BD+DA = B(A+B).

We can now factor a "D" from the left side and get D(B+A) = B(A+B).

Thus, D(B+A) = B(A+B), and as B+A = A+B, we can divide both sides by A+B and get B=D, as desired.