Methods for showing that 4 points lie on the same circle? I'm looking for commonly used methods in contest geometry to show that 4 point lie on the same circle. Are there any tricks besides using the fact that the angles across add up to 180°?
 A: If the coordinates of the four different
points are known, then they are
concyclic points if and only
if the following determinant is zero.
$$0 = \begin{vmatrix}
x_1^2+y_1^2&x_1&y_1&1\\
x_2^2+y_2^2&x_2&y_2&1\\
x_3^2+y_3^2&x_3&y_3&1\\
x_4^2+y_4^2&x_4&y_4&1
\end{vmatrix}$$
This is a consequence of the general equation of a circle being
$$ 0 = (x^2 + y^2) + ax + by + c. $$
A: We can prove that quadrilateral $ABCD$ is cyclic by infinitely many methods:

*

*If there is a point $O$ in the plane of the quadrilateral $ABCD$ such that $OA=OB=OC=OD$, so $ABCD$ is cyclic;

*If midperpendiculars to sides of quadrilateral $ABCD$ are intersected in the same point, so $ABCD$ is cyclic;

*If for quadrilateral $ABCD$ we have $\measuredangle ABD=\measuredangle ACD,$ so $ABCD$ is cyclic;

*Let for quadrilateral $ABCD$ we have $AB\cap CD=\{P\},$ $B$ and $C$ be midpoints of $PQ$ and $PR$ respectively, $AM$ and $DM$ be perpendiculars to $AB$ and $CD$ respectively. Now,  if $M\in QR$, so $ABCD$ is cyclic;
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A: Let the points be considered as lying on the complex plane, say, $z_1,z_2,z_3,z_4$. Then, consider the map (also called the cross ratio of $z,z_2,z_3,z_4$) given by,
$$
M(z)=\left(\frac{z-z_3}{z-z_4}\right)/\left(\frac{z_2-z_3}{z_2-z_4}\right).
$$
This is a Mobius transformation which sends $z_2,z_3,z_4$ to $1,0,\infty$ respectively. Since Mobius transformations preserve circles (including straight lines considered as circles), it maps the circle made by $z_2,z_3,z_4$ to the real axis. Also, Mobius transformations are invertible, hence, no other point which is not on the above circle will be mapped to the real axis.
Hence, if $M(z_1)$ is real, $z_1$ lies on the circle formed by $z_2,z_3,z_4$.
In conclusion,
if
$$
\left(\frac{z_1-z_3}{z_1-z_4}\right)/\left(\frac{z_2-z_3}{z_2-z_4}\right)
$$
is real, then $z_1,z_2,z_3,z_4$ lie on a circle.
