# Reference request that explains the logic and ideas behind auxiliary constructions in synthetic geometry

I'll be frank. I'm really, really bad at synthetic geometry problems that require auxiliary constructions before they can be solved. I just don't know which lines I should draw to reduce the problem to angle chasing or something as simple.

At the moment, constructions are a really hit-or-miss thing for me. Just keep drawing lines and eventually something will click, right? Well it might work for some problems, but as they get harder, I don't think it'll work anymore.

When I look at the solutions for the problems, I can understand what is going on, and how the construction makes it simpler, but what I can't understand is how or why someone would think of constructing those lines. As a result, I can't apply the logic or method of thinking elsewhere.

I've tried to classify the logic with each problem I solve into a set of principles. For example, if I see a right angled triangle, I try drawing a median from the right angle vertex, to see if that simplifies the problem by introducing isoceles triangles by the converse of Thales' Theorem. But this is just proving to be a really tedious, and honestly not very reliable or effective, process.

In short I'd like a reference where the author describes constructions through example problems in an almost algorithmic or systematic way. Or at least manages to help better my judgement in what constructions to make, as well as allowing me to see why those constructions would simplify things in the short run, if not the whole problem. I'm fine with any kind of reference: textbooks, online notes, blogs, or websites. I would prefer it if the reference were freely available though, like a free pdf.

• When I look at the solutions for the problems ...you need to realize that what you see is the auxiliary construction which does work for that problem. In all but trivial cases, the solver probably tried several other constructs, sometimes many other constructs, which did not work. Do not expect to always and easily see the right auxiliary construct upfront. Rather, try anything that looks promising, and learn to quickly recognize whether that might help. – dxiv Aug 7 '16 at 19:49
• That's exactly the problem with such solutions: the train of thoughts the author of the solution went through to find it is not supplied with the solution, so it is usually impossible to recover it and apply for solving other similar problems. And this is very bad. – BarbaraKwarc Aug 19 '16 at 22:51