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Is the following statement true or false?

The following figure shows a triangle $A B C$ , where $A B = A C .$ $D$ and $E$ are points on $A B$ and $A C$ such that $A D = A E$ , the points $A, B, C$ and $D$ are concyclic.

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True

False

answer is A.

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Detailed Solution

Given that

A B = A C

D

and

E

are points on

A B

and

A C

such that

A D = A E

and the points

A, B, C

and

D

are concyclic.
Question Image

We show this using the converse of the Thales theorem. When a transversal intersects two lines, the two lines are parallel only if the interior angles and exterior angles on the same side of the transversal are supplementary.

AD = AE

…..i)

AB = A C

…..ii)
Subtracting

AD

from both sides of the equation,

⇒ A B − A D = A C − A D ∵ A D = A E ⇒ A B − A D = A C − A E

BD = EC…. iii)
Dividing equation i) by equation iii),
We get,

A D B D = A E E C

Now using the converse of Thales' theorem we get

D E ∥ B C

.
The sum of interior angles is

180 °

.
Therefore,

⇒ ∠ DEC + ∠ ECB = 180 °

Since

AB = AC ⇒ ∠ B = ∠ C

⇒ ∠ D E C + ∠ C B D = 180 °

The quadrilateral BCED is cyclic.
Hence B, C, E, D are concyclic points which is true.
Therefore the correct option will be 1.

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