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°?

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.
