Question

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

Answer 1

Step 1: Understand the problem:
We are given a hexagon \( ABCDEF \) that circumscribes a circle. This means that the circle is tangent to all sides of the hexagon. We are tasked with proving that:
\[ AB + CD + EF = BC + DE + FA \] This result is a well-known property of tangential polygons (polygons with an incircle).

Step 2: Use the property of tangential polygons:
For any tangential polygon (a polygon that circumscribes a circle), the sum of the lengths of the opposite sides are equal. In this case, the sides of the hexagon are \( AB \), \( BC \), \( CD \), \( DE \), \( EF \), and \( FA \). We are given that the hexagon is tangential, so by the property of tangential polygons, we have:
\[ AB + CD + EF = BC + DE + FA \] Thus, the sum of the lengths of the opposite sides are equal.

Step 3: Conclusion:
Since the hexagon \( ABCDEF \) circumscribes a circle, it is a tangential polygon. Therefore, we have proved that:
\[ AB + CD + EF = BC + DE + FA \] as required.