# AIMO 2015 Q10

Geometry Level 5

$$X$$ is a point inside an equilateral triangle $$ABC$$. $$Y$$ is the foot of the perpendicular from $$X$$ to $$AC$$, $$Z$$ is the foot of the perpendicular from $$X$$ to $$AB$$, and $$W$$ is the foot of the perpendicular from $$X$$ to $$BC$$.

The ratio of the distances of $$X$$ from the three sides of the triangle (respectively, $$AC$$, $$AB$$ and $$BC$$) is $$1 : 2 : 4$$.

If the area of $$AZXY$$ is $$13 \text{ cm}^2$$, find the area of $$ABC$$.

Investigation

If $$XY:XZ:XW = a:b:c$$, find the ratio of the areas of $$AZXY$$ and $$ABC$$. Write your answers with proof in the solutions.

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