Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.7 AIME, harder Prove it Philippines

Problem:

The figure shows a semicircle, where $B$ is a point on its diameter $AC$ so that $AB = 6$ and $BC = 12$ . The line perpendicular to this diameter meets the semicircle at $D$ . The semicircles with diameters $AB$ and $BC$ , drawn as shown, meet $AD$ and $CD$ at $E$ and $F$ , respectively. Find the distance between $E$ and $F$ .

Solution

Solution:

Observe that $DEFB$ is a rectangle so $EF = DB$ . However, $DB^2 = AB \times BC = 72$ . Thus, $EF = 6\sqrt{2}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.