Prove circle similarity and recover a circle equation

Try

Two circular signs have different sizes and centers. A motion alone can move their centers but cannot change the radius. A positive dilation can resize one, but scaling about an arbitrary origin usually moves its center to the wrong place. A correct similarity must handle both placement and radius, and its image must fill the entire target circle.

For a fixed example, map the circle centered at C=(1,−2) with radius 3 onto the circle centered at D=(−4,1) with radius 6. Translate C to the origin, dilate by 2, then translate to D. The combined point rule is F(x,y)=(2x−6,2y+5). It sends the source rightmost point (4,−2) to the target rightmost point (2,1).

Predict

Predict what happens to any source point, not only the rightmost one. Could a proposed mapping send every sampled point to the target but miss some unsampled arc? Then consider (x−2)²+(y+1)²=0 and the same left side equal to −9. Decide whether both can be called circles under the positive-radius definition.

Experiment

Use the fixed rule on four source points and verify their distances from D. Include the point (1,1), whose image is (−4,7), as well as the two left/bottom extrema. Keep equal axis units so the displayed Euclidean geometry agrees with the calculation.

Source point	Image under F	Distance from D=(−4,1)
(4,−2)	(2,1)	6
(1,1)	(−4,7)	6
(−2,−2)	(−10,1)	6
(1,−5)	(−4,−5)	6

Now reverse the procedure for an arbitrary target point Y: subtract D, divide its displacement by 2 and add C. Verify algebraically that applying F returns Y. This inverse action supplies evidence beyond a handful of samples.

Understand

For arbitrary centers C,D and radii r,s>0, define F(X)=D+(s/r)(X−C). If X lies on the source circle, then its image has distance (s/r)r=s from D, so the image is contained in the target. Conversely, for every Y at distance s from D, set X=C+(r/s)(Y−D). This X is distance r from C and F(X)=Y. Thus the image equals the target locus. The map is a translation, a positive dilation and another translation, so it is a similarity. Both positive-radius assumptions are needed for the inverse divisions.

A circle centered at (h,k) with radius r>0 also has a precise coordinate description. The distance from (x,y) to its center is √((x−h)²+(y−k)²), by the independently established Pythagorean distance formula. Equality to r implies (x−h)²+(y−k)²=r². Conversely any real solution has distance √(r²)=r, since the radius is positive. Squaring is reversible here because both sides of the distance equality are nonnegative; that sign check must be stated rather than assumed in every squared equation.

To recover a circle from an expanded equation, complete the two squares. For example, x²+y²+6x−4y−12=0 becomes (x+3)²+(y−2)²=25. The center is (−3,2), radius 5. The linear coefficients determine the center but the constant determines whether a positive radius remains. If the original constant were +13 instead, the right side would be zero, giving the single point (−3,2). If it were +17, the right side would be −4, giving no real points.

These degeneracies are geometrically meaningful. A single point can be called a zero-radius degeneration if the convention is explicit, but it is not covered by the positive-circle similarity theorem. A negative squared radius is not an imaginary drawing on the real coordinate plane; it means the real locus is empty. Unequal coefficients on x² and y² also require analysis: they do not generally describe a circle just because squared variables appear.

Apply

Given a purported circle equation, recover its center and squared radius before choosing a construction. Verify one point using the original distance condition and explain the full reverse implication. Given two circles, provide the complete similarity rule and its inverse, rather than only a scale factor. The center calculation and the onto argument answer different parts of the question.

Continue

Next use similarity to justify how circular arcs scale, then derive radian and sector-area formulas. The same principle will connect a finite construction to a general measurement rule, with the limiting step stated explicitly.