Which result is true?

Geometry Level 4

An equilateral triangle $ABC$ of side length $l$ is inscribed in a circle and let $P$ be a point on the circumcircle. Is it true that $PA^2+PB^2+PC^2=l^2$ ?

Is true for a specific point No Yes

1 solution

Sathvik Acharya
Apr 9, 2017

The correct result is $PA^2+PB^2+PC^2=2l^2$

Hint: Let $P$ and $A$ lie on opposite sides of side $BC$ . Use Ptolemy's theorem on quadrilateral $ABPC$ to get $PA=PB+PC$ . Then use the Cosine rule on $\triangle APC$ and $\triangle APB$ to get the desired result.

Which result is true?

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