ইন্টুইশন: বিবর্ধন কোণের মান সংরক্ষণ করে

Given a center $O$ and a scale factor $k$, a dilation is a transformation that maps every point $P$ to the point $p$ such that $\overrightarrow{Op}=k\cdot\overrightarrow{OP}$.

From this, can we show intuitively that $\angle abc\cong\angle ABC$ for any such dilation? (In the diagram below, the points $P,A,B,C$ are dilated about $O$ with two scale factors: $k=1.5$ sends the points to $p,a,b,c$, and $k=-0.7$ sends the points to $q,x,y,z$. We'll focus on $k=1.5$, but the same reasoning applies to $k=-0.7$.)

From pure intuition, and from perhaps trying some examples, it seems as though a dilation maps a figure to something that is essentially the same figure, just not the same size. All of the properties that do not depend on size are the same. Thus, a natural way to proceed is to show that any triangle is mapped to a similar triangle.

Clearly $\triangle AOB\sim\triangle aOb$; of course, the symmetric results also hold.

Thus, we have that $ab=k\cdot AB$, as well as the symmetric results.

This means that $\triangle abc\sim\triangle ABC$, and therefore $\angle abc\cong\angle ABC$.

We have thus intuitively shown that dilations preserve angles.