ABC is an isosceles triangle in which AB = AC. If D and E are midpoints of AB and AC respectively, prove that the points D, B, C, E are concyclic.

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Solution

ΔABC is an isosceles triangle in which AB = AC and D and E are the midpoints of AB and AC, respectively.
By mid point theorem, we have:
DE parallel to BC
⇒ ∠ADE = ∠ABC ...(i)
Also, AB = AC
⇒ ∠ABC = ∠ACB ...(ii)
From (i) and (ii), we have:
∠ADE = ∠ACB
Now, ∠ADE + ∠EDB = 180° [∵ ADB is a straight line]
⇒ ∠ACB + ∠EDB = 180°
Since, Opposite angles of a quadrilateral are supplementary
Hence, DBCE is a cyclic quadrilateral or the points D, B, C and E are concyclic.

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