Exercise 1945Geometry › Plane geometryL1Added 4 Nov 2026

Squares on two sides and the median

#complex numbers#squares#median#perpendicularity

Statement

Let $ABC$ be a triangle. Squares $ABDE$ and $ACFG$ are erected outwards on the sides $AB$ and $AC$ , with $E$ adjacent to $A$ on the first and $G$ adjacent to $A$ on the second. Let $M$ be the midpoint of $BC$ . Prove that $EG=2\,AM$ and $EG\perp AM$ .

Hint

Put $A$ at the origin of the complex plane and orient the triangle counter-clockwise: $B=b$ , $C=c$ . Rotations by $\pm90^\circ$ are multiplications by $\pm i$ .

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Solution

Idea. The outer vertices are $-ib$ and $ic$ ; their difference is $i(b+c)=2i\,M$ .

The vertices. The interior lies to the left of $\overrightarrow{AB}$ , so the outward square gives $E=-ib$ . The interior lies to the right of $\overrightarrow{AC}$ , so the outward square gives $G=ic$ .

The computation. $M=\frac{b+c}2$ and
$G-E=i(b+c)=2i\,M.$
Multiplication by $i$ is rotation by $90^\circ$ with $|i|=1$ , so $EG=2AM$ and $EG\perp AM$ . $\blacksquare$

Remark. $[AEG]=[ABC]$ ; the median of $ABC$ is the altitude of $AEG$ .

Lesson. A square is a $90^\circ$ rotation; with $A$ as origin, such configurations reduce to one-line computations with $i$ .

Squares on two sides and the median

Statement

Solution

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