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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 & ECD
BC=ED (sides of a regular polygon are equal)
CD=CD (common side)
Therefore,
Triangle BCD is congruent to triangle ECD
Therefore,
Since these angles are equal, it proves the theorem "Equal chords of a circle subtend equal angles".
Thus, vertices B , C , D , E lie on a circle and are hence CONCYCLIC.
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∙ 12y ago
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What shape has more than four vertices?
There are infinitely many possible answers: a pentagon, hexagon, heptagon, etc.
How does a picture of a pentagon look like?
A picture of a pentagon looks like a house--that is a five sided figure with four vertices. Other objects that you may be familiar with that are pentagons are: a baseball home plate, the Pentagon building in DC, or a soccer ball patch.
How many vertices does a trapezoid have?
A trapezium is a quadrilateral and so has four sides and four vertices.
How many vertices does a pyramid have?
Vertices in a PyramidA tetrahedron is the simplest pyramid. It has four vertices and four faces.A pyramid, the same general shape as monumental structures in Egypt, with a square base and four triangular faces meeting at an apex has 5 vertices.However, a pyramid can have any polygon as its base. A pyramid with an n-sided polygon as base would have n+1 vertices.So a twelve sided pyramid would have 13 vertices I think there are 4......* * * * *One more than the number of vertices (corners) on the base. A triangular pyramid has 3+1 = 4 vertices, a rectangular pyramid has 4+1 = 5 and so on.It depends. There are different kinds of pyramids, with different amounts of vertices (the plural for vertex). There are square pyramids, with square bases, triangle pyramids, with triangular bases, hexagon pyramids, octagon pyramids, pentagon pyramids... here are a few answers, though: Square pyramids have five vertices. Triangle pyramids have four. Hexagon pyramids have seven, octagon pyramids have nine, pentagon pyramids have six.6
How many faces edges vertices of tetrahedron?
Four faces, six edges and four vertices.
What shape has more than four vertices?
There are infinitely many possible answers: a pentagon, hexagon, heptagon, etc.
How does a picture of a pentagon look like?
A picture of a pentagon looks like a house--that is a five sided figure with four vertices. Other objects that you may be familiar with that are pentagons are: a baseball home plate, the Pentagon building in DC, or a soccer ball patch.
What has 6 vertices and 10 edges and one face is a rectangular pentagon?
There can be no answer to this question. This is because there cannot be a face that is a rectangular pentagon. Rectangular means it has four straight sides while pentagon means it has five straight sides. It can have either 4 or 5 sides - NOT both!
How many vertices does a trapezoid have?
A trapezium is a quadrilateral and so has four sides and four vertices.
How many vertices are on a square?
Four vertices Four equal sides Four equal angles
How many verticals in a rectangle?
There are 4
How much faces and vertices does a square pyramid have?
Four faces four vertices
How many vertices does a pyramid have?
Vertices in a PyramidA tetrahedron is the simplest pyramid. It has four vertices and four faces.A pyramid, the same general shape as monumental structures in Egypt, with a square base and four triangular faces meeting at an apex has 5 vertices.However, a pyramid can have any polygon as its base. A pyramid with an n-sided polygon as base would have n+1 vertices.So a twelve sided pyramid would have 13 vertices I think there are 4......* * * * *One more than the number of vertices (corners) on the base. A triangular pyramid has 3+1 = 4 vertices, a rectangular pyramid has 4+1 = 5 and so on.It depends. There are different kinds of pyramids, with different amounts of vertices (the plural for vertex). There are square pyramids, with square bases, triangle pyramids, with triangular bases, hexagon pyramids, octagon pyramids, pentagon pyramids... here are a few answers, though: Square pyramids have five vertices. Triangle pyramids have four. Hexagon pyramids have seven, octagon pyramids have nine, pentagon pyramids have six.6
How many vertices's are there in four trapezoid?
16 vertices.
How many vertices are in a trapezium?
Four vertices are in a trapezium.
Does the pentagon has four sides?
No, a pentagon has 5 sides.
Which shape has four vertices and four faces?
A triangular based pyramid has four vertices and four faces
