Question

If a hexagon ABCDEF circumscribes a circle, prove that AB + CD + EF = BC + DE + FA.

Answer 1

Step 1: Problem Statement:
A hexagon \( ABCDEF \) is circumscribed by a circle, meaning the circle is tangent to each side. The objective is to prove the equality: \( AB + CD + EF = BC + DE + FA \). This is a known characteristic of tangential polygons.

Step 2: Applying Tangential Polygon Property:
For any polygon that circumscribes a circle (a tangential polygon), the sum of alternating sides are equal. For hexagon \( ABCDEF \), this property directly yields the required equation:
\[AB + CD + EF = BC + DE + FA\]

Step 3: Proof Confirmation:
As hexagon \( ABCDEF \) circumscribes a circle, it is a tangential polygon. Therefore, the equality \( AB + CD + EF = BC + DE + FA \) is proven.