Question 8

Circumscribed Quadrilateral: A quadrilateral that has a circle inside it touching all four sides. The circle is called the inscribed circle (incircle). Each side of the quadrilateral is a tangent to the circle.

Tangents from an External Point are Equal: If two tangents are drawn from the same external point to a circle, they are always equal in length. Each vertex of ABCD acts as an external point with two tangents going to the circle — this is the key property the entire proof rests on.

Points of Tangency (P, Q, R, S): The points where the circle touches each side of the quadrilateral. Each side has exactly one such point.

We will use the property that tangents from an external point to a circle are equal in length.

Step 1 — List equal tangents

Let's look at the given diagram. The circle is inscribed in the quadrilateral ABCD. Points P, Q, R, S are the points of tangency. Each vertex of the quadrilateral is an external point. Tangents drawn from point A are AP and AS. Tangents drawn from point B are BP and BQ. Tangents drawn from point C are CR and CQ. Tangents drawn from point D are DR and DS. So, we can write these equalities:

$AP = AS \quad \text{(1)}$

$BP = BQ \quad \text{(2)}$

$CR = CQ \quad \text{(3)}$

$DR = DS \quad \text{(4)}$

Step 2 — Add the equations

Let's add all four equations together. We add the left sides. We add the right sides.

$AP + BP + CR + DR = AS + BQ + CQ + DS$

Step 3 — Group terms to form sides

Now, let's rearrange the terms. We will group them to form the sides of the quadrilateral. On the left side, we group $(AP + BP)$ and $(CR + DR)$. On the right side, we group $(AS + DS)$ and $(BQ + CQ)$.

$(AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)$

We know that $AP + BP$ forms the side AB. We know that $CR + DR$ forms the side CD. We know that $AS + DS$ forms the side AD. We know that $BQ + CQ$ forms the side BC. Substituting these into the equation:

$AB + CD = AD + BC$

This proves the required statement.

Answer

$\boxed{\text{AB} + \text{CD} = \text{AD} + \text{BC}}$