Methods for showing that 4 points lie on the same circle?

Question

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°?

Answer 1

We can prove that quadrilateral $ABCD$ is cyclic by infinitely many methods:

1. If there is a point $O$ in the plane of the quadrilateral $ABCD$ such that $OA=OB=OC=OD$, so $ABCD$ is cyclic;
2. If midperpendiculars to sides of quadrilateral $ABCD$ are intersected in the same point, so $ABCD$ is cyclic;
3. If for quadrilateral $ABCD$ we have $\measuredangle ABD=\measuredangle ACD,$ so $ABCD$ is cyclic;
4. 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; $$\bullet$$ $$\bullet$$ $$\bullet$$

Answer 2

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 $$ 0 = (x^2 + y^2) + ax + by + c. $$

Answer 3

Here is a way to tell from the distances between the points without knowing their rotational order.

Leg $d_{ij}$ be the distance between two points $i$ and $j$. By Ptolemy's Theore, if the rotational order of the points is 1-2-3-4 then

$d_{13}d_{24}=d_{12}d_{34}+d_{14}d_{23}; d_{13}d_{24}-d_{12}d_{34}-d_{14}d_{23}=0$

For the other two possible rotational orders we get similar results:

$d_{12}d_{34}-d_{14}d_{23}-d_{13}d_{24}=0$

$d_{14}d_{23}-d_{13}d_{24}-d_{12}d_{44}=0$

Now take all three cases into account with one blow by multiplying the three zero factors together. It looks inelegant, true --- unless we slip in an extra factor:

$d_{13}d_{24}+d_{12}d_{34}+d_{14}d_{23}=0$

The product of a four factor equations is then a simple form that involves only even powers:

$(d_{12}d_{34})^4+(d_{13}d_{24})^4+(d_{14}d_{23})^4-2(d_{12}d_{34})^2(d_{13}d_{24})^2-2(d_{13}d_{24})^2(d_{14}d_{23})^2-2(d_{14}d_{23})^2(d_{12}d_{34})^2=0$

$\color{blue}{(d_{12}d_{34})^4+(d_{13}d_{24})^4+(d_{14}d_{23})^4=2[(d_{12}d_{34})^2(d_{13}d_{24})^2+(d_{13}d_{24})^2(d_{14}d_{23})^2+(d_{14}d_{23})^2(d_{12}d_{34})^2]}$

With this formulation, you do not need square root extractions to get the even powers of the distances in the above expression. You will, however, run into large numbers with seemingly small coordinates. For instance, consider the four points

$1\to(0,5);2\to(5,0);3\to(3,4);4\to(3,-4).$ (They're "rigged" to lie on $x^2+y^2=5^2$).

Then from the Pythagorean formula for squared distance

$d_{12}^2=50,d_{13}^2=10,d_{14}^2=90,d_{23}^2=20,d_{24}^2=20,d_{34}^2=64$

And then

$(d_{12}d_{34})^4+(d_{13}d_{24})^4+(d_{14}d_{23})^4=3200^2+200^2+1800^2=\color{blue}{13520000}$

$2[(d_{12}d_{34})^2(d_{13}d_{24})^2+(d_{13}d_{24})^2(d_{14}d_{23})^2+(d_{14}d_{23})^2(d_{12}d_{34})^2]=2×[(3200×200)+(3200×1800)+(200×1800)]=\color{blue}{13520000}.$

Answer 4

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.