Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Prove it Italy

Problem:

Let $ABC$ be a right triangle at $A$ , with $\angle ABC = 15^\circ$ . Let $H$ be the foot of the altitude from $A$ and let $J, K$ be the projections of $H$ onto $AB$ and onto $AC$ . Knowing that the area of $AJHK$ is $45~\mathrm{cm}^2$ , how many $\mathrm{cm}^2$ is the product $BJ \cdot CK$ ?

Solution

Solution:

The answer is $45$ . Setting $x = AJ$ , $y = AK$ , $x' = JB$ and $y' = KC$ , using the second Euclid's theorem we have $x \cdot x' = y^2$ and $y \cdot y' = x^2$ . Multiplying side by side we get $\left(x \cdot x'\right)\left(y \cdot y'\right) = \left(x' \cdot y'\right)(x \cdot y) = x^2 \cdot y^2$ , from which $x' \cdot y' = x \cdot y = 45$ .

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.