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Statement

Theorem 1 (p. 293). There are positive constants $c_1, c_2$ with

$$c_1 n^2\log n < f_2^i(n) < c_2 n^{5/2}.$$

Notation. For distinct points $X_1,\ldots,X_n$ in $k$-dimensional Euclidean space $E_k$, the paper writes $f_k^i(n)$, $f_k^e(n)$, $f_k^c(n)$ and $f_k^s(n)$ for the largest possible number of isosceles triangles (congruent or not), of equilateral triangles, of pairwise congruent triangles and of pairwise similar triangles among them, the maximum taken over all choices of the $n$ points (pp. 291–292). Here $c$, $c_1$, $c_2$ are positive constants, not necessarily the same at each occurrence (p. 291).

Source. P. Erdős and G. B. Purdy, Some extremal problems in geometry, III, Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, 1975), Congress. Numer. XIV, Utilitas Math., Winnipeg, 1975, pp. 291–308. The edition read is identified on the source card. Theorem 1 on p. 293, proof pp. 293–295.

Read depth. Claims checked: the statement was read clause by clause on the printed page.

Proof pointer

Upper bound (pp. 293–294): fix a vertex; for each other point the apexes of isosceles triangles on that base lie on a line (the perpendicular bisector), and these lines are distinct. Two lines share at most one point, so a dyadic count of the lines by their number of points bounds the triangles with a given base vertex by $cn^{3/2}$. Lower bound (pp. 294–295): the integer points of a square grid of side about $\sqrt n$; a central grid point is the apex of $\binom{r(k)}{2}$ isosceles triangles on the circle of radius $\sqrt k$ about it, where $r(k)$ counts representations of $k$ as a sum of two squares, and the mean value of $r^2(k)$ (Ramanujan; Hardy and Wright) gives $cn\log n$ triangles per vertex for $cn$ vertices.

Bears on

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