I'll give you the problem first (from here), so you can run off and try to solve it before reading on. By all means, draw your own diagram as well, based only on the description provided. The extensive labeling in the diagram here is provided mostly as a key for the explanation below.
In \(\small\Delta\mathtt{ABC}\), points \(\small\mathtt{D}\) and \(\small\mathtt{E}\) are interior points of \(\small\mathtt{\overline{BC}}\) and \(\small\mathtt{\overline{AC}}\), respectively. \(\small\mathtt{\overline{BF}}\) bisects \(\small\measuredangle\mathtt{CBE}\), and \(\small\mathtt{\overline{AF}}\) bisects \(\small\measuredangle\mathtt{DAC}\). Prove that \(\small\mathtt{m\measuredangle AEB} + \small\mathtt{m\measuredangle ADB} = \small\mathtt{2 \times m\measuredangle AFB}\).
Mixing and Matching Letters
The first thing I started doing here was to map Euclid's I.32 wherever I could. It turned out that I didn't really have to reverse course after this. I saw various triangles as having angle measure sums of 180\(\small\mathtt{^\circ}\), and that was a good enough start. So, I noted for example, the following:
\(\small\Delta\mathtt{ABD}\): \(\small\mathtt{d + 2y + b + a = 180^\circ}\)
\(\small\Delta\mathtt{AFB}\): \(\small\mathtt{x + a + b + y + f = 180^\circ}\)
\(\small\Delta\mathtt{AEB}\): \(\small\mathtt{2x + a + b + e = 180^\circ}\)
And we are looking to show that \(\small\mathtt{e + d = 2f}\). From the first two equations, I can work out that \(\small\mathtt{d = 2x + c}\), and from the first and last equations, I can work out that \(\small\mathtt{e = 2y + c}\). Finally, from the first and third equations, I've got \(\small\mathtt{f = x + y + c}\).
We see that \(\small\mathtt{2y + c}\) + \(\small\mathtt{2x + c}\) = \(\small\mathtt{2x + 2y + 2c}\) = \(\small\mathtt{2(x + y + c)}\), which is in fact \(\small\mathtt{e + d = 2f}\), or \(\small\mathtt{m\measuredangle AEB} + \small\mathtt{m\measuredangle ADB} = \small\mathtt{2 \times m\measuredangle AFB}\).
Proximity Is Positively Correlated with Mundaneness
It's worth noting what I used to work on this proof: the Angle Sum Theorem, some algebra (properties of equality, I guess), and one little bit of "insight." I recognized that when I was able to write four equations with all of the important variables and a lot of overlapping terms, I was probably going to be able to isolate what I wanted to know. In other words, I could trace every advantage I had in completing this proof to just long-term memory and practice.
We sometimes get a little too worked up in mathematics education in particular. Of course, no, this isn't even close to being a difficult proof for most high school math teachers, I expect, but I also think if those teachers' tenth-graders were routinely working on proofs like this independently, they would have the sense that the future was going to be okay. And in order for students to do this, nothing more would be required, in general, than remembering stuff and practicing. Keep calm.