In the given figure, if $A B = A C,$ prove that $B E = E C .$
OR
$A B C$ is an isosceles triangle in which $A B = A C,$ circumscribed about a circle, as shown in the given Figure. Prove that the base is bisected by the point of contact.

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Solution

Since tangents from an exterior point to a circle are equal
in length.
$A D = A F [T a n g e n t s f r o m A] - - - - (i)$
$B D = B E [T a n g e n t s f r o m B] - - - - (i i)$
$C E = C F [T a n g e n t s f r o m C] - - - - (i i i)$

Now,
$A B = A C$ [Given]
$\Rightarrow A B - A D = A C - A D$ $[$ Subtracting $A D$ from both sides $]$
$\Rightarrow A B - A D = A C - A F$ $[$ using $(i)$ $]$
$\Rightarrow B D = C F$
$\Rightarrow B E = C F$ $[$ using $(i i)$ $]$
$\Rightarrow B E = C E$ $[$ using $(i i i)$ $]$

Hence Proved

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Similar questions

In the given figure, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that the point of contact P bisects the base BC.

Q. In the given figure, an isosceles triangle ABC with AB = AC, circumscribes a circle. Prove that the point of contact P bisects the base BC
[CBSE 2012]

Q. In the figure, an isosceles triangle

A B C,

with

A B = A C,

circumscribes a circle. Prove that the point of contact

P

bisects the base

B C .

Prove that the lengths of tangents drawn from an external point to a circle are equal. Using the above do the following:

ABC is an isosceles triangle in which AB = AC, circumscribed about a circle as shown in the fig. Prove that the base is bisected by the point of contact.

A B C

is an isosceles triangle, in which

A B = A C

, circumscribed about a circle. Show that

B C

is bisected at the point of contact ?

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In the given figure, if AB=AC, prove that BE=EC.ORABC is an isosceles triangle in which AB=AC, circumscribed about a circle, as shown in the given Figure. Prove that the base is bisected by the point of contact.

In the given figure, if $A B = A C,$ prove that $B E = E C .$
OR
$A B C$ is an isosceles triangle in which $A B = A C,$ circumscribed about a circle, as shown in the given Figure. Prove that the base is bisected by the point of contact.