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Golden Triangle

Age 16 to 18
Challenge Level

triangle ABC.

The three triangles $ABC$, $CBD$ and $ABD$ are all isosceles. Find the angles in the triangles.

The sides $AB$ and $BC$ have lengths $p$ and $q$ respectively. Prove that the ratio $p/q$ is equal to the golden ratio $\frac{1}{2} (\sqrt{5}\ +1) $.

and find the ratio $q/p$.

The area of triangle $ABC$ is 2 square units. Find the areas of $CBD$ and $ABD$ exactly (i.e. find the areas in the form \[ a + b \sqrt{5} \]

where $a$ and $b$ are rational numbers)