Inscribed Equilateral Triangle

Geometry Level 4

Let $\triangle{ABC}$ be an equilateral triangle inscribed in a circle, and let $D$ be a point on the arc $BC$ that $A$ does not lie on. If $BD=3$ and $CD=4$ , find $AD$ .

The answer is 7.

1 solution

Victor Loh
Jun 21, 2015

Using Ptolemy's Theorem on cyclic quadrilateral $ABDC$ , we have

$AB \times CD + AC \times BD = AD \times BC.$

Since $\triangle{ABC}$ is equilateral, $AB=AC=BC$ . Substituting $BD=3$ and $CD=4$ into the above equation,

$AB \times 4 + AB \times 3 = AD \times AB \implies \boxed{AB = 7}.$

I used a more synthetic approach to solve . But a nice soln and nice question

Aditya Kumar - 5 years ago

Can you share your synthetic approach?

Shaun Leong - 4 years, 9 months ago

AD should be 7 ;)

Peter van der Linden - 4 years, 9 months ago

Inscribed Equilateral Triangle

The answer is 7.

1 solution

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