Fancy point

Author: anonymous
Problem has been solved: 11 times

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Triangle $ABC$ is given, in which $\angle A = 30^\circ, \angle B = 45^\circ, \angle C = 105^\circ$. Let $O, H, I, L$ be the circumcenter, orthocenter, incenter, and the intersection point of $ABC$'s symmedians, respectively. Let $T = OL \cap HI$ and $y, z$ be the distances from $T$ to the lines $CA, AB$, respectively. It is known that $y + z = 1$. Let $P(x)$ be the minimal polynomial of number $y$, coefficients of which are setwise coprime integers (the leading coefficient is greater than 0). Find $P(-1)$. WolframAlpha can help you with it. The minimal polynomial is needed only for checking your answer.