In space, there are four distinct points, $A, B, C$, and $D$. We know that there is an isometry of space that swaps $A$ and $B$, and leaves $C$ and $D$ fixed; furthermore, there is an isometry that maps $A$ to $B$, $B$ to $C$, $C$ to $D$, and $D$ to $A$. What is the measure of angle $ABC$? And what is the measure of angle $DAC$?
Congruence transformations are distance-preserving, meaning the length of any segment is equal to the length of its image. In the first transformation, the image of segment $AC$ is $BC$, and the image of segment $AD$ is $BD$, so $AC = BC$ and $AD = BD$. In the second transformation, the image of $AB$ is $BC$, the image of $BC$ is $CD$, and the image of $CD$ is $DA$, so $AB = BC = CD = DA$. Together with the previous results, this means that the distance between any two of the points $A, B, C$, and $D$ is the same. This is only possible if the points are the vertices of a regular tetrahedron.

![](https://cdn.mathpix.com/cropped/2024_05_02_41db6417979fbb523577g-1.jpg?height=436&width=569&top_left_y=408&top_left_x=756)

# Figure 1

In the case of a regular tetrahedron, the transformations described in the problem indeed exist. The first transformation can be, for example, a reflection over the plane that contains points $C$ and $D$ and the midpoint of segment $AB$ (Figure 1). The second transformation can be a $120^{\circ}$ rotation around the line connecting the centroid of the $ABD$ face with point $C$, followed by a reflection over the plane that contains the (new) points $A$ and $B$ and the midpoint of segment $DC$ (Figures 2 and 3).

![](https://cdn.mathpix.com/cropped/2024_05_02_41db6417979fbb523577g-1.jpg?height=360&width=504&top_left_y=1175&top_left_x=797)

Figure 2

![](https://cdn.mathpix.com/cropped/2024_05_02_41db6417979fbb523577g-1.jpg?height=349&width=518&top_left_y=1704&top_left_x=782)

Figure 3

Each face of a regular tetrahedron is an equilateral triangle, so both angles mentioned in the problem are $60^{\circ}$.

Based on the work of Gábor Veres (Balassagyarmat, Balassi B. Gymnasium, 1st year).