How to get the coordinates inside

[sarah22]

What's the formula on how to get the coordinates inside of:
a) Square
b) Triangle
c) Circle
d) Oval

I tried the formula here, for triangle, (http://2000clicks.com/mathhelp/GeometryPointAndTriangle3.htm ) but sometimes even though the point is inside it says it is not.

 

[yyat]

A half-plane is given by an equation of the form $$xn_x+yn_y-d\ge 0$$ where $$(n_x,n_y)$$ is the inward pointing normal vector of the boundary line, and $$d=p_xn_x+p_yn_y$$, where $$(p_x,p_y)$$ is a point on the boundary.

To test if a point is inside the half-plane, you only need to check the above inequality.

Since a triangle (resp. a square) is the intersection of 3 (resp. 4) half-planes, you just need to figure out their equations and do the test for each one.

For an ellipse of the form $$ax^2+by^2-c=0$$, the corresponding inequality for the points inside it is $$ax^2+by^2-c\le 0$$.