Prove that any four vertices of a regular pentagon are concyclic (lie on the same circle).
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Question

Prove that any four vertices of a regular pentagon are concyclic (lie on the same circle).

KnowledgeBoat · Accepted Answer

Let ABCDE be a regular pentagon. ∴ ∠ABC = ∠BCD = ∠CDE = ∠DEA = ∠EAB. Interior angle of a regular polygon is given by : So, each interior angle of a regular pentagon = = 108°. In △AED, AE = ED [Sides of a regular pentagon are equal] ∴ ∠EAD = ∠EDA [Angles opposite to equal sides are equal] .........(1) ⇒ ∠AED + ∠EAD + ∠EDA = 180° ⇒ 108° + ∠EAD + ∠EAD = 180° ⇒ 2∠EAD = 180° - 108° ⇒ 2∠EAD = 72° ⇒ ∠EAD = ⇒ ∠EAD = 36°. ∴ ∠EDA = 36°. From figure, ∠BAD = ∠BAE - ∠EAD = 108° - 36° = 72°. In quadrilateral ABCD, ∠BAD + ∠BCD = 72° + 108° = 180°. Since, sum of opposite angles = 180°, which is possible when the quadrilateral is cyclic quadrilateral. Hence, proved that ABCD is a cyclic quadrilateral.

Mathematics

Prove that any four vertices of a regular pentagon are concyclic (lie on the same circle).

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