Basic Rules of Cyclic Quadrilaterals

$ABCD$ is cyclic if and only if:

or

If $ABCD$ is cyclic, then

Conversely, suppose

Assume on the contrary that $ABCD$ is not cyclic. Draw a circle through $A,B,$ and $D$.

$ABC'D$ is cyclic, so

We also know

Therefore,

But $\angle C'$ is the exterior angle of $\triangle BC'C$, so

which gives a contradiction.

$ABCD$ is cyclic if and only if:

If $ABCD$ is cyclic, then

Conversely, suppose

Assume on the contrary that $ABCD$ is not cyclic. Draw a circle through $A,B,$ and $D$.

$ABC'D$ is cyclic, so

We know

So

which gives a contradiction.

$ABCD$ is cyclic if and only if:

Assume $ABCD$ is cyclic.

By AA,

So

Conversely, assume

Then

By SAS,

Therefore,

Hence $ABCD$ is cyclic.

$ABCD$ is cyclic if and only if:

Assume $ABCD$ is cyclic.

Therefore,

Similarly,

By AA,

So

Conversely, assume

Then

By SAS,

Therefore,

Hence $ABCD$ is cyclic.

In an acute triangle $ABC$, draw the three altitudes. Find six cyclic quadrilaterals in the resulting configuration.

Four circles are arranged so that each circle is tangent to its two adjacent circles, as shown. Let $A,B,C,D$ be the four points of tangency. Prove that $ABCD$ is cyclic.

Two circles intersect at $M$ and $N$, and $P$ lies between $M$ and $N$. Two lines through $P$ intersect the first circle at $B,D$ and the second circle at $A,C$, respectively. Prove that $ABCD$ is cyclic.

Two circles intersect at $M$ and $N$, and $P$ lies on the extension of $MN$. Two lines through $P$ intersect the first circle at $A,D$ and the second circle at $B,C$, respectively. Prove that $ABCD$ is cyclic.

Two circles intersect at $M$ and $N$, and $AB$ is their line of centers. The lines $MN,FC,$ and $ED$ are concurrent at $P$. Prove that $ABCD$ is cyclic.