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Source: Imre Leader, Paul A. Russell and Mark Walters, Transitive sets and cyclic quadrilaterals, Journal of Combinatorics 2 (2011), no. 3, 457--462: the definitions and projection remark opening Section 2 on p. 459, the sphericity remark on p. 457 and the cyclicity remark on p. 458. The edition read is identified on the source card. The paper states these facts as remarks, with at most a one-line reason; the proofs below are written here.

Statement

A quadrilateral is a set of four coplanar points, with coincidences allowed; it is trivial when all four points coincide. A quadrilateral $xyzw$ has parameters $\alpha,\beta$ when

$$w=z+\alpha(x-z)+\beta(y-z). \tag{1}$$

For a quadrilateral in a vector space $V\oplus W$, the projections onto $V$ and onto $W$ satisfy (1) with the same parameters, and if the quadrilateral is nontrivial, at least one of the two projections is nontrivial.

Every finite transitive set is spherical. Consequently, four distinct coplanar points that embed in a finite transitive set lie on a circle.

Full proof

Equation (1) is an affine relation: its coefficients on $x,y,z$ are $\alpha,\beta,1-\alpha-\beta$ and sum to one. Linear projection therefore preserves it. If the projections onto both $V$ and $W$ were trivial, the four points would have the same $V$-coordinate and the same $W$-coordinate, so the original quadrilateral would be trivial. Repeating this argument over a finite orthogonal direct-sum decomposition gives the corresponding assertion for any number of summands.

For the spherical assertion, let $T$ be a finite transitive set and let

$$c=\frac1{|T|}\sum_{t\in T}t$$

be its centroid. Every isometry preserving $T$ permutes its points and hence fixes $c$. Transitivity therefore makes $\lVert t-c\rVert$ independent of $t\in T$, so $T$ lies on one sphere centered at $c$.

If four distinct points of $T$ are coplanar, their plane meets that sphere in a circle: the intersection cannot be empty or a single tangent point because it contains four distinct points. Thus those four points are cyclic. This proves the source's observation that cyclicity is automatic for a distinct quadrilateral embedded in a finite transitive set.

When $x,y,z$ are noncollinear, the parameters in (1) are unique. In particular, three distinct points on a circle are noncollinear, since a line meets a circle in at most two points. This is the setting used in Theorem 1.

Used by. Lemma 4, Theorem 1, and the almost-every consequence.

Bears on. Problem 174: the sphericity of finite transitive sets is the reason every subtransitive set is spherical, so that the subtransitive sets form a subclass of the spherical ones; the rest of the page is a tool for Theorem 1.