Equilateral makes it easy

Geometry Level pending

A point $P$ lies on the circumcircle of an equilateral $\triangle ABC$ such that $P$ lies on the arc $BC$ , $BP=4$ and $CP=6$ .Find $AP$ ?

The answer is 10.

1 solution

Deepak Kumar
Jan 23, 2016

Hint:Think about use of Ptolmey's identity for a cyclic quadrilateral which states that sum of product of opposite sides=product of diagonals.One of the diagonals is the side of the given equilateral triangle given here which can be found using cosine rule here and other diagonal is AP itself.

I think there is no need to find the side of the $\triangle{ABC}$ ,as

$AP*BC=AC*BP+AB*CP$ and since the $\triangle{ABC}$ is equilateral $AB=BC=AC$

$AP=BP+CP$

Siddharth Singh - 5 years, 4 months ago

Equilateral makes it easy

The answer is 10.

1 solution

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