1. Layering and Essence of Transformations
Definition: Homothetic Figures If two figures are not only similar but also all lines connecting corresponding points pass through the same point, then the figures are called homothetic, and this point is known as the homothety center.
Congruent figures are a special case of similarity with a ratio of 1. Translation, reflection, and rotation preserve congruence; homothety changes size through scaling while maintaining shape.
2. Core Constraint from Similarity to Homothety
Similar figures require equal corresponding angles and proportional corresponding sides; homothetic figures add the strong constraint that all lines joining corresponding points must pass through a single point.
Property: Characteristics of Homothetic Figures
1. All homothetic figures are similar, but not all similar figures are homothetic.
2. The ratio of distances from corresponding points to the homothety center equals the similarity ratio.
3. Dimensional Leap: The Square Law of Area
Understand how the ratio of side lengths (similarity ratio $k$) affects higher-order properties: perimeter ratio follows $k$, and area ratio follows $k^2$. This intrinsic rule becomes especially intuitive in homothety transformations.
If a poster measuring $10 \text{ cm} \times 5 \text{ cm}$ has its side lengths enlarged by 3 times, although the perimeter increases only by 3 times, the physical area covered increases by a factor of $3^2 = 9$.