For example we have a circle and triangle. We need to check if at given $(m, n)$ coordinate triangle is intersecting with circle (area).
Circle center is fixed at $(100, 100)$ with radius $R = 50$. Triangle corners coordinates vary: $$ A(x_1, y_1)\quad B(x_2, y_2)\quad C(x_3, y_3). $$
Circle equation is: $(x - 100)^2 + (y - 100)^2 = 50^2$. If $(m - 100)^2 + (n - 100)^2 - 50^2$ is below or equal zero, it means that $(m, n)$ coordinate is inside circle.
Now I want to extend this to one/simple formula, which on a simple more/less/equal condition would evaluate if $(m, n)$ is inside both circle and triangle. Is this possible? What triangle formula should I use and what condition should I look for?
You might ask why I want to construct simple formula. Well, I am working on a problem, where I want to brutally scan certain coordinates region and to each coordinate assign number from which it would be obvious if this coordinate is both inside triangle and circle or not.
I have read (triangle case) that I should check barycentric coordinates for equality greater than zero. I did understand how it works, just not sure how could I simplify everything into one equation.