Melon-and-bean (spiral similarity)

When to use

• Two triangles sharing a vertex with unequal leg lengths — one weaker hypothesis than [[hand-in-hand]] (which requires isosceles)
• The problem says "rotate by $\alpha$ about fixed point $O$ while scaling by $k$" — this is the canonical language of spiral similarity
• A moving point $Q$ yields a moving point $P$ via "rotate + scale" about a fixed point $O$; find the locus of $P$ (same shape as $Q$'s)
• The half-angle model on a non-isosceles triangle ([[half-angle]] requires $AB = AD$; without that, congruence degrades to similarity)
• The rotational view of "K-shape similarity" ([[one-line-three-equal-angles]])

Core move

Shared vertex + shared apex angle + matching leg ratios = a similar pair of "hands". Rotation angle $=$ shared apex angle $\alpha$; scaling ratio $k =$ ratio of adjacent leg lengths; the locus shape of $Q$ determines the locus shape of $P$ ("plant melons, harvest melons; plant beans, harvest beans" — guā dòu in Chinese).

Construction

Let $\triangle OAB \sim \triangle OCD$ (sharing vertex $O$, $\angle AOB = \angle COD = \alpha$, $\dfrac{OA}{OB} = \dfrac{OC}{OD}$), with scale $k = \dfrac{OC}{OA} = \dfrac{OD}{OB}$:

1. Common-angle addition / subtraction: $\angle AOC = \angle AOB + \angle BOC = \angle BOC + \angle COD = \angle BOD$
2. Rearrange the ratio: from $\dfrac{OA}{OB} = \dfrac{OC}{OD}$ get $\dfrac{OA}{OC} = \dfrac{OB}{OD}$ ("swap means and extremes")
3. SAS similarity: $\triangle OAC \sim \triangle OBD$ (matching ratio of sides + matching angle), with ratio $\dfrac{OB}{OA}$
4. Read the conclusion:
   • "Hand"-segment ratio: $\dfrac{BD}{AC} = \dfrac{OB}{OA}$ (no longer equal lengths, but proportional)
   • Angle between "hands": the angle between $BD$ and $AC$ equals $\alpha$ (same as the shared apex angle — consistent with [[hand-in-hand]])

Why it works

The spiral similarity "rotate by $\alpha$ about $O$, scale by $k$" is a composite transformation: rotation + dilation. It sends $A \to C$ and simultaneously $B \to D$. The transformation preserves:

• Angles: the angle between any two segments is unchanged
• Ratios: the ratio of any two segment lengths is unchanged

By composition, the angle between $\overrightarrow{AC}$ and $\overrightarrow{BD}$ is always $\alpha$ and the length ratio is always $k$.

This is the geometric essence of the "melon-and-bean principle": after one spiral similarity, the locus of $P$ has the same shape as the locus of $Q$ — "plant melons, harvest melons; plant beans, harvest beans". Concretely:

• $Q$ on a line $\ell$ $\Rightarrow$ $P$ on a line $\ell'$ (the image of $\ell$ under the spiral similarity)
• $Q$ on a circle $\odot M$ $\Rightarrow$ $P$ on a circle $\odot M'$ (radius scaled by $k$, centre is the image of $M$)

Worked examples

• Right triangles $\triangle OAB$ ($\angle O = 90°$) and $\triangle OCD$ ($\angle O = 90°$) share vertex $O$, with $\dfrac{OA}{OB} = \dfrac{OC}{OD}$; show $AC \perp BD$ and $\dfrac{BD}{AC} = \dfrac{OB}{OA}$ (the shared-apex $90°$ special case)
• Non-isosceles half-angle variant: a non-isosceles triangle with apex $2\alpha$ ($AB \neq AD$); $\angle EAF = \alpha$ ($E \in BC$, $F \in CD$); the relation between $EF$ and $BE + DF$ is no longer equality but a weighted equality with a similarity ratio

• Locus problem: $Q$ slides on a line $\ell$; $P$ is obtained from $Q$ by rotating $60°$ about $O$ and scaling by $\dfrac{1}{2}$; find the locus of $P$ (answer: another line $\ell'$)

Variants / generalizations

• Degenerate to congruence ($k = 1$): similarity ratio is $1$, the spiral similarity reduces to "pure rotation" — i.e. the [[hand-in-hand]] model; the angle formula $\alpha$ is unchanged
• Degenerate to dilation ($\alpha = 0$): zero rotation, pure scaling — reduces to a dilation centred at $O$; corresponding-point connectors are concurrent at $O$
• Non-isosceles version of the half-angle model: [[half-angle]]'s "two polyline pieces splice into a single straight segment" holds in the isosceles case; without isosceles, the splice is a polyline weighted by the similarity ratio, which this model handles
• Auto-detection of the rotation centre: given two corresponding segments $AB \to CD$, the centre $O$ of the spiral similarity is uniquely determined by the geometric condition that $\triangle OAC$ and $\triangle OBD$ must simultaneously be similar ("four points concyclic + line intersection" construction)
• Locus principle (the "melon-and-bean"): the geometric basis for "locus analysis" in moving-point optimization — in essence, the middle-school version of "linear transformations preserve shape"