# Pi and triangles? No!

Geometry Level 4

A pentagon has respective side lengths $$a$$, $$a \sqrt 3$$, $$a$$, $$a$$, and $$a$$, and angles $$90^\circ$$, $$90^\circ$$, $$120^\circ$$, $$120^\circ$$, and $$120^\circ$$, both corresponding. This pentagon can be split into an equilateral triangle and two other triangles.

Find the smallest positive integer of $$a$$ which satisfies the condition the product of the areas of the three triangles divided by their sum is greater than $$100000 \pi$$.

×