Inverse arc under an inversion

Problem statement

Question 1.1 (2.5 marks; you answer one of the two questions, 1.1 or 1.2). Determine the inverse arc of $AB$ under the inversion that transforms circle $c$ into circle $c'$. Explain the basis of the construction used.

Figure in the statement: circle $c$, with centre $C$, lies inside the larger circle $c'$, and the two are internally tangent at a point $P$ on the line of centres. Point $B$ is the end of $c$ opposite $P$, and $A$ is the point of $c$ on the perpendicular to that line through $C$. Arc $AB$, drawn with a thick line, is the quarter circle from $A$ to $B$.

Data read from the figure (the statement does not write them): the centre of $c'$ coincides with $B$ and the radius of $c'$ is twice that of $c$. We take $r=[[r]]$ drawing units for $c$ and $r'=[[rp]]$ for $c'$.

Source: Madrid university entrance exam (PAU, Spain) · 2025-2026 · Ordinary session · Technical Drawing · Question 1.1. Official exam set by the public universities of the Community of Madrid, to whom it belongs; translated from Spanish and reproduced with its source in order to explain it for teaching purposes.