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What kind of polygon has all vertices lie on a circle?
It is a regular polygon as for example an equilateral triangle
Prove four vertices of pentagon are concyclic?
First make sure you understand that concyclic simply means the points lie on a common circle. We are not told it is a regular pentagon but we will assume it is. We could create pentagons that are not even convex and would not be concyclic.Let's start with a regular pentagon. You can split it up into 5 congruent triangles with the points meeting at the middle. Any side of one of these triangle is connected from each of the vertices of the pentagon to the center of the pentagon. Since all 5 triangles are congruent, this distance must be the same for each of the vertices. So, we see that each of the vertices is equidistant from a given point. Now if we drew a circle centered at that point with a radius equal to the distance between the point and any vertex, that circle would pass through all 5 vertices. Therefore any four ( really all 5), vertices of a regular pentagon are concyclic.A nice proof would use Ptolemy's theorem. I will place a line to an answers.com page that helps with that.Another solution:The pentagon has to be regular. Otherwise, the question is impossible to prove.Consider a regular polygon ABCDE.Take triangles BCD & ECDBC=ED (sides of a regular polygon are equal)CD=CD (common side)
What is a polygon where all diagonals lie on its interior?
A convex polygon.
What is the fewest vertices a polyhedron can have?
Four, as in a tetrahedron. Any three points lie in a plane, so if you only had three vertices they couldn't make a three-dimensional object.
What is a non-degenerate triangle?
A non-degenerative triangle has 3 vertices which does not lie on a stright line. i.e the are not collinear
