Recently, I have been playing with symmetry tools and I found (in my sample space) that a unique point exists with an attribute to a fixed angle made on each vertex.
A clearer view:
Consider a triangle $ABC$. Through each vertex of the triangle, a line passes such that it makes an angle $\alpha$ with the adjacent side. I found that, if $$\tan\alpha= \frac{4\Delta}{a^2+b^2+c^2}$$ all the additional lines intersect at one and only one point.
I reverse engineered the last equation to get $\alpha$; I considered that they necessarily intersect at a point $O$, then divided the triangle into 3 sections such that cosine rule and area summation are applied taking only $cos\alpha$ and $sin\alpha$ respectively; Then just divide $\frac{\sin\alpha}{\cos\alpha}$ to get the result
This point of triple intersection such lines inclined to a definite angle $\alpha$ is called a Brocard point.
$$\text{My Question}$$
- If I were to iteratively repeat the same process for a random angle alpha (subject to make a triangle inside $ABC$), will it converge to a single point, (assuming it does) will that point be different from the Brocard point as obtained primarily?
links to relevant pages: