Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.5 AIME Prove it Austria

Let $ABC$ be an acute triangle and $D$ be a point on the altitude through $C$ .
Prove that the mid-points of the line segments $AD$ , $BD$ , $BC$ and $AC$ form a rectangle.

Solution

We denote with $M_{XY}$ the mid-point of the line segment $XY$ .
Using the intercept theorem, we deduce that
* $M_{AD}M_{BD}$ is parallel to $AB$ .
* $M_{AC}M_{BC}$ is parallel to $AB$ .
* $M_{AC}M_{AD}$ is parallel to $CD$ .
* $M_{BC}M_{BD}$ is parallel to $CD$ .
Therefore, $M_{AD}M_{BD}$ is parallel to $M_{AC}M_{BC}$ and $M_{AC}M_{AD}$ is parallel to $M_{BC}M_{BD}$ .
Furthermore, $M_{AC}M_{BC}$ is orthogonal to $M_{AC}M_{AD}$ , since $CD$ is orthogonal to $AB$ .
Therefore, the four mid-points form a rectangle. $\square$

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