Question

Prove that any four vertices of a regular pentagon lie on a circle.​

Answer 1

Answer:

To prove: Every set of four vertices of ABCDE is a set of points lying on a circle. ... ✴️ Thus, AB subtends equal angle at two points C and E on the same side of AB. ✴️ Therefore, the points A, B, C, E are concyclic. ✴️ Similarly, every set of four vertices of pentagon ABCDE is a set of concyclic points

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Answer 2

Answer:

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> Given: A regular pentagon ABCDE.

> To prove: Every set of four vertices of ABCDE is a set of points lying on a circle.

> Proof:

✴ First we show that the points A, B, C, E lie on a circle.

✴ Join AC and BE.

✴ In ∆ABC and ∆BAE, we have:

✴ AB = BA (common)

✴ BC = AE (sides of a regular pentagon)

✴ angle ABC = angle BAE (each equal to 108°)

✴ Therefore, ∆ABC is congruent to ∆BAE (by SAS congruency rule)

✴ => angle BCA = angle AEB (by C.P.C.T.)

✴ Thus, AB subtends equal angle at two points C and E on the same side of AB.

✴ Therefore, the points A, B, C, E are concyclic

✴ Similarly, every set of four vertices of pentagon ABCDE is a set of concyclic points.

✨……Hence, PROVED……✨

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