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Tangential Quadrilaterals

In this explanation, we explain some properties of tangential quadrilaterals and give a necessary and sufficient condition for a quadrilateral to be tangential.

What are they?

A tangential quadrilateral is defined as a quadrilateral that has an inscribed circle. Notice that not all quadrilaterals have this property; for example, draw a circle tangent to three of its sides and then draw the fourth side to not touch the circle. Then, the quadrilateral is hence is not tangential.

Properties

Let us take a diagram of a tangential quadrilateral and its inscribed circle:

Note that $AS=AP$, $BP=BQ$, etc., by equal tangents. Then, we have: $$AB+CD = a + b + c + d = AD + BC = s$$ Thus, in a tangential quadrilateral, the sums of opposite sides are equal and equal the semiperimeter. However, this is a necessary condition for being a tangential quadrilateral, and not a sufficient one. To prove the converse, we will use contradiction:

Assume the opposite. Then, we have non-tangential quadrilateral $ABCD$ such that $AB + CD = AD + BC$. We draw the circle tangent to $AB, BC, CD$. Then, we draw the second tangent from $A$ to the circle, and extend it to hit $CD$ at point $D'$. By the condition proven above, we have: $$AB + CD' = AD' + BC$$ Subtracting this from the given equation, we have: $$DD' = CD - CD' = AD - AD'$$ $$DD' + AD' = AD$$ This violates the triangle inequality, so we have a contradiction. $\Box$

So therefore, a quadrilateral is tangential if and only if the sums of the opposite sides are equal.

That's it for tangential quadrilaterals. Thanks for reading!

Tangential Quadrilaterals

In this post we are going to talk about tangential quadrilaterals.

Lets learn how to tell if a quadrilateral has an inscribed circle!

Pitot's Theorem states that:

In a tangential quadrilateral the two sums of lengths of opposite sides are the same.