Is Your Point Inside a Non-Regular Tetrahedron?

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To determine if a point lies inside a non-regular tetrahedron, one effective method is to calculate the volumes of sub-tetrahedra formed with the point and the tetrahedron's vertices. If the sum of these sub-tetrahedral volumes equals the volume of the original tetrahedron, the point is inside. This approach is analogous to checking point inclusion in a triangle by comparing areas. Additional resources and formulas for volume calculations can be found online. This method provides a reliable solution for the problem presented.
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Hello ,
does anybody knows how can I check if a point is inside a tetrahedral. The tetrahedral isn't regular.

Thanks!
 
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I don't know if this is a good way to solve it. But if in a triangle, the easiest way to check if a point is inside is to check the sum of the area. i.e. for ABC, there is a point P, if P is inside ABC, then, area(ABP)+area(ACP)+area(BCP) = area(ABC).

You can check the volume in the tetrahedral case.
 
Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?

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