Geometry Problem 502
Triangle, Two Squares, Midpoint, Perpendicular, Half the Measure
Problem Statement
Let $ABDE$ and $BCGH$ be squares constructed externally on sides $AB$ and $BC$ of triangle $ABC$, respectively. Let $M$ be the midpoint of $DH$.
To Prove:
Strategic Synthetic Hints
- Idea 1 — Rotation: Consider a rotation of $90^\circ$ about point $B$. How does it map square $ABDE$ to a congruent position, and what does it imply about segment $AC$ and the line $BM$?
- Idea 2 — Midpoint and Congruence: Construct point $N$ as the midpoint of $AC$. Show that $\triangle BDE$ is congruent to some triangle involving $N$. What happens to the segments $BM$ and $AC$?
- Idea 3 — Vectors / Coordinate: Place $B$ at the origin, $A$ and $C$ as vectors $\mathbf{a}$ and $\mathbf{c}$. Express the coordinates of $D$ and $H$ in terms of rotations. Compute the midpoint $M$ and verify both perpendicularity and length relation.
- Idea 4 — Double Square Configuration: The two squares share vertex $B$. Relate the segment $DH$ to the diagonal $AC$ via the midpoint theorem in a suitably constructed parallelogram or triangle.
Foundation Theorems — Direct Tools
Pattern note: The two-square configuration in Problem 1622 builds on the same rotational idea, applied cyclically to all vertices. The midpoint relation in Problem 502 is the key lemma for the hexagonal sum.
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