Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Prove it Canada

Problem:

Let $ABC$ be an acute-angled triangle. Inscribe a rectangle $DEFG$ in this triangle so that $D$ is on $AB$ , $E$ is on $AC$ and both $F$ and $G$ are on $BC$ . Describe the locus of (i.e., the curve occupied by) the intersections of the diagonals of all possible rectangles $DEFG$ .

Solution

Solution:

The locus is the line segment joining the midpoint $M$ of $BC$ to the midpoint $K$ of the altitude $AH$ . Note that a segment $DE$ with $D$ on $AB$ and $E$ on $AC$ determines an inscribed rectangle; the midpoint $F$ of $DE$ lies on the median $AM$ , while the midpoint of the perpendicular from $F$ to $BC$ is the centre of the rectangle. This lies on the median $MK$ of the triangle $AMH$ .

Conversely, any point $P$ on $MK$ is the centre of a rectangle with base along $BC$ whose height is double the distance from $K$ to $BC$ .

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