* Problem* : If a, b and c are the lengths of the sides of a triangle

*ABC*and if \( p_1 , p_2 \) and \( p_3 \) are the lengths of the perpendiculars drawn from the circumcentre onto the sides BC, CA and AB respectively, then show that

.

* Solution* : Let O be the circum centre

= 2

Similarly,

&

Now, =

= =

**Lemma**

For &

(lemma proved)

So,

= [ as ]

=

=

( proved )