# Iron man mark 1 (43 to go) series of challenges for geniuses

Geometry Level 4

Suppose a line 'l' cuts an equilateral triangle $$ABC$$ of side $$\sqrt{3}$$ in two points different from the vertices. Say it cuts $$AB$$ and $$AC$$ in points $$R$$ and $$Q$$ respectively. We mark the orthocentre $$H$$ of the triangle $$ARQ$$ and also the midpoint $$M$$ of side $$RQ$$. We extend $$HM$$ to a point $$T$$ so that $$HM = MT$$. The point $$P$$ is the foot of perpendicular from $$T$$ on side $$BC$$.

Now, we draw outward equilateral triangles $$RA'Q$$, $$QC'P$$, and $$PB'R$$. (The Napoleonic triangles of triangle $$PQR$$).

The task is to find: $$A'T + B'T + C'T$$.

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