Is there an easy way to find the sign of this determinant without calculating it directly?











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There exist real numbers $A_x, A_y, B_x, B_y, C_x, C_y, D_x$ and $D_y$. Is there an easy way to find the sign of following determinant without calculating it directly? BTW, the determinant appears when judging whether or not point $D$ is inside the circumcircle of triangle $ABC$ on the plane.



$$
left|
begin{array}{ccc}
A_x - D_x & A_y - D_y & (A_x - D_x)^2 + (A_y - D_y)^2 \
B_x - D_x & B_y - D_y & (B_x - D_x)^2 + (B_y - D_y)^2 \
C_x - D_x & C_y - D_y & (C_x - D_x)^2 + (C_y - D_y)^2
end{array}
right|
$$










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  • At a glance, this would be way easier to read if the subscripts were removed and every argument was just assigned a single letter.
    – The Count
    yesterday















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There exist real numbers $A_x, A_y, B_x, B_y, C_x, C_y, D_x$ and $D_y$. Is there an easy way to find the sign of following determinant without calculating it directly? BTW, the determinant appears when judging whether or not point $D$ is inside the circumcircle of triangle $ABC$ on the plane.



$$
left|
begin{array}{ccc}
A_x - D_x & A_y - D_y & (A_x - D_x)^2 + (A_y - D_y)^2 \
B_x - D_x & B_y - D_y & (B_x - D_x)^2 + (B_y - D_y)^2 \
C_x - D_x & C_y - D_y & (C_x - D_x)^2 + (C_y - D_y)^2
end{array}
right|
$$










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cia_rana is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • At a glance, this would be way easier to read if the subscripts were removed and every argument was just assigned a single letter.
    – The Count
    yesterday













up vote
0
down vote

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up vote
0
down vote

favorite











There exist real numbers $A_x, A_y, B_x, B_y, C_x, C_y, D_x$ and $D_y$. Is there an easy way to find the sign of following determinant without calculating it directly? BTW, the determinant appears when judging whether or not point $D$ is inside the circumcircle of triangle $ABC$ on the plane.



$$
left|
begin{array}{ccc}
A_x - D_x & A_y - D_y & (A_x - D_x)^2 + (A_y - D_y)^2 \
B_x - D_x & B_y - D_y & (B_x - D_x)^2 + (B_y - D_y)^2 \
C_x - D_x & C_y - D_y & (C_x - D_x)^2 + (C_y - D_y)^2
end{array}
right|
$$










share|cite|improve this question









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cia_rana is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











There exist real numbers $A_x, A_y, B_x, B_y, C_x, C_y, D_x$ and $D_y$. Is there an easy way to find the sign of following determinant without calculating it directly? BTW, the determinant appears when judging whether or not point $D$ is inside the circumcircle of triangle $ABC$ on the plane.



$$
left|
begin{array}{ccc}
A_x - D_x & A_y - D_y & (A_x - D_x)^2 + (A_y - D_y)^2 \
B_x - D_x & B_y - D_y & (B_x - D_x)^2 + (B_y - D_y)^2 \
C_x - D_x & C_y - D_y & (C_x - D_x)^2 + (C_y - D_y)^2
end{array}
right|
$$







determinant triangle circle triangulation






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edited yesterday





















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asked Nov 15 at 18:29









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cia_rana is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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Check out our Code of Conduct.












  • At a glance, this would be way easier to read if the subscripts were removed and every argument was just assigned a single letter.
    – The Count
    yesterday


















  • At a glance, this would be way easier to read if the subscripts were removed and every argument was just assigned a single letter.
    – The Count
    yesterday
















At a glance, this would be way easier to read if the subscripts were removed and every argument was just assigned a single letter.
– The Count
yesterday




At a glance, this would be way easier to read if the subscripts were removed and every argument was just assigned a single letter.
– The Count
yesterday















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