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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