Cyclic and Tangential Quadrilaterals

Try this: cyclic quadrilaterals

A quadrilateral whose four vertices all lie on one circle is called a cyclic quadrilateral.

With $D$ outside the circle, $∠ B + ∠ D$ is less than $18 0^{\circ}$ . With $D$ inside, it is more than $18 0^{\circ}$ .
Click “Snap D onto the circle through A, B, C”: both sums of opposite angles become $18 0^{\circ}$ at the same moment.
With $D$ on the circle, drag it along the arc (when you let go it snaps back onto the circle): $∠ D$ does not change, because inscribed angles on the same arc are equal, and $∠ B + ∠ D$ stays $18 0^{\circ}$ .
Now drag $A$ or $C$ to change the circle, and snap $D$ back on: the result still holds.

Predict first

Quadrilateral $A B C D$ is inscribed in a circle and $∠ A = 7 0^{\circ}$ . How many degrees is $∠ C$ ?

Choose an answer first; then you can see the result.

Opposite angles of a cyclic quadrilateral are supplementary

Step 1 / 5

Four concyclic points
$A$ , $B$ , $C$ , $D$ all lie on the dashed circle (the circle light shows ✓).
∠B stands on one arc
∠D stands on the rest
Add them
An exterior angle equals the opposite interior angle

The converse is true too: if a quadrilateral has supplementary opposite angles, its four vertices are concyclic. This is one of the most common ways to prove that four points are concyclic (the “Concyclic Points” stop in the Circles volume goes further).

Tangential quadrilaterals

A quadrilateral whose four sides all touch one circle is called a tangential quadrilateral. The dotted circle below is tangent to the lines of the three sides $A B$ , $B C$ , $D A$ . The question is whether it also touches the fourth side $C D$ .

The solid pair of opposite sides is

A B

C D

; the dashed pair is

B C

D A

. Drag

D

until the two pairs have equal sums.

Tangential quadrilateral: AB + CD = BC + DA

Step 1 / 4

All four sides are tangent
This kite has an incircle (the light shows ✓). Let the points of tangency on the sides $A B$ , $B C$ , $C D$ , $D A$ be $P$ , $Q$ , $R$ , $S$ .
The tangent-length theorem
Split each side in two
The same four numbers

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: $∠ B + ∠ D = 18 0^{\circ}$

$D$ is locked onto the circle. Drag it along the arc, or drag $A$ , $B$ , $C$ ( $D$ follows and stays on the new circle).

Practice

Exercise 1Basic

In the cyclic quadrilateral $A B C D$ , $∠ A : ∠ B : ∠ C = 2 : 3 : 7$ . Find $∠ D$ in degrees.

Exercise 2Basic

Quadrilateral $A B C D$ is circumscribed about a circle, with $A B = 5$ , $B C = 7$ , $C D = 8$ . Find the length of $D A$ .

Exercise 3Advanced

Quadrilateral $A B C D$ is inscribed in $⊙ O$ , $E$ is on the extension of $B C$ , and $∠ D C E = 7 5^{\circ}$ . Find the central angle $∠ B O D$ in degrees (the central angle on the minor arc $B D$ ).

Exercise 4Challenge

An isosceles trapezoid has an incircle, an upper base of $2$ and a lower base of $8$ . Find the radius of the incircle.

Try this: cyclic quadrilaterals

Opposite angles of a cyclic quadrilateral are supplementary

Tangential quadrilaterals

Tangential quadrilateral: AB + CD = BC + DA

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Remembering “opposite angles supplementary” as “opposite angles equal”

Remembering “equal sums of opposite sides” as “equal opposite sides”

Cyclic and Tangential Quadrilaterals

Try this: cyclic quadrilaterals

Why: two inscribed angles share the whole circle

Tangential quadrilaterals

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop