Center of Dilation
The center of dilation is the fixed point from which all points of a figure are scaled outward or inward during a dilation.
Definition
The center of dilation is the one fixed point that does not move when a dilation is performed; all other points move toward or away from it, and lines connecting original points to their images all pass through it. Formally, the center $C$ satisfies $D(C,k)(C) = C$ for any scale factor $k$; for any point $P$, the image $P'$ lies on ray $CP$ (or its opposite ray if $k < 0$) at distance $|k| \cdot |CP|$ from $C$. Solving $f(x) = kx + (1-k)C = x$ shows $x = C$ is the unique fixed point of the dilation map; for a composition of two dilations with different centers and $k_1 k_2 \neq 1$, the new center is found by intersecting the lines through corresponding image pairs.
Example
Dilating a triangle from the origin by scale factor $3$ moves every vertex to a point $3$ times farther out along the line from the origin through that vertex. To find the center of dilation between triangle $ABC$ and its image $A'B'C'$, draw lines $AA'$ and $BB'$; the center is where they intersect, and $CC'$ should pass through the same point. Composing $D(C_1=0, k_1=2)$ and $D(C_2=(6,0), k_2=1/3)$ gives composed scale factor $2(1/3) = 2/3$, with new center $x = (9,0)/2 = (4.5, 0)$.
Key Insight
Zoom in on a photo using a pinch gesture: the point you pinch from is the center of dilation, with everything expanding or contracting from that fixed point. The center can be inside, outside, or on the original figure, and if it sits at a vertex, that vertex maps to itself; it also plays the role of the "vanishing point" in perspective drawing, where parallel lines appear to meet. In projective geometry, the center of dilation corresponds to the "center of perspectivity" in Desargues' theorem, with dilation as the affine special case of this projective concept.