Sum of interior angles of cyclic hexagon

Click For Summary
SUMMARY

The sum of the interior angles at vertices A, C, and E of a cyclic hexagon equals the sum of the interior angles at vertices B, D, and F. This property arises from the inscribed angle theorem, which states that angles subtended by the same arc are equal. For cyclic polygons, this relationship generalizes, indicating that the sum of interior angles at alternate vertices remains consistent across all cyclic shapes.

PREREQUISITES
  • Understanding of cyclic polygons and their properties
  • Familiarity with the inscribed angle theorem
  • Basic knowledge of polygon interior angles
  • Ability to apply geometric proofs
NEXT STEPS
  • Research the inscribed angle theorem in depth
  • Explore properties of cyclic quadrilaterals and their angle relationships
  • Study the generalization of angle sums in cyclic polygons
  • Practice geometric proofs involving cyclic shapes
USEFUL FOR

Mathematicians, geometry students, educators, and anyone interested in advanced geometric properties and proofs related to cyclic polygons.

Natasha1
Messages
494
Reaction score
9
Right I have been given the following problem and cannot resolve it. I have had an attempt but without much success. Could anyone help me with this exercise, please? Hints or a little more welcome :-)

A cyclic hexagon is a hexagon whose vertices all lie on the circumference of a circle.

The vertices of a cyclic hexagon are labelled in order A to F. Prove that the sum of the interior angles at A, C and E is equal to the sum of the interior angles at B, D and F.

Generalise (concisely) to other cyclic polygons?
 
Physics news on Phys.org
what have you done so far?
 

Similar threads

Proving Angle Sums in Cyclic Polygons: A Generalization Approach
  • · Replies 1 ·
Replies
1
Views
2K
Proving Cyclic Quadrilateral: Opposite Angles Sum 180°
  • · Replies 2 ·
Replies
2
Views
2K
(Rigorously) Prove that two medians of a triangle intersect
  • · Replies 2 ·
Replies
2
Views
3K
Angles inscribed in circles part 1
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Sum of a polygon's interior angles
  • · Replies 39 ·
2
Replies
39
Views
23K
Undergrad Solve Quadrilateral Exterior Angles Sum Equal to Interior Angles Sum
  • · Replies 5 ·
Replies
5
Views
2K
Magnetic field outside a conducting hollow cylinder
  • · Replies 8 ·
Replies
8
Views
5K
How Do You Prove a Sum of Cubes is Theta of n^4 Using Inequalities?
  • · Replies 1 ·
Replies
1
Views
7K
Graduate Proving Angle Parallelograms in Convex Polygons of 5+ Sides
  • · Replies 14 ·
Replies
14
Views
4K
Graduate Lines on a circle - a counting problem
  • · Replies 10 ·
Replies
10
Views
3K