Two Strange Constructions in the Euclidean Plane

We present two new constructions in the usual euclidean plane. We only deal with ‘Grecian Geometry’, with this phrase we mean elementary geometry in the two-dimensional space R 2 . We describe and prove two propositions about ‘projections’. The proofs need only elementary analytical knowledge.

💡 Research Summary

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The paper under review claims to present two “strange” constructions in the Euclidean plane, but in reality it merely restates elementary affine‑geometric facts using a mixture of analytic and synthetic language. The author works in ℝ² with the usual Cartesian axes and introduces four main propositions together with two auxiliary lemmas.

Proposition 1
Two distinct parallel lines (G_S) and (G_T) are given, together with a third line (L) that meets each of them at points (S) and (T) respectively and does not pass through the origin (O=(0,0)). The lines through the origin and the intersection points are denoted (Z_S) and (Z_T). Two cases are distinguished:

Case (A) – Neither (G_S) nor (G_T) is horizontal. Let (a_S) and (a_T) be the x‑intercepts of (G_S) and (G_T). There exists a unique point (P_{\text{hor}}=(x_{\text{hor}},y_{\text{hor}})\in L) such that the translated point ((x_{\text{hor}}-a_S,,y_{\text{hor}})) lies on (Z_T) and ((x_{\text{hor}}-a_T,,y_{\text{hor}})) lies on (Z_S).

Case (B) – Neither (G_S) nor (G_T) is vertical. Let (b_S) and (b_T) be the y‑intercepts. There exists a unique point (P_{\text{ver}}=(x_{\text{ver}},y_{\text{ver}})\in L) such that ((x_{\text{ver}},,y_{\text{ver}}-b_S)\in Z_T) and ((x_{\text{ver}},,y_{\text{ver}}-b_T)\in Z_S).

The proof proceeds by parametrising (L) as ((x_S,y_S)+\tau (w_1,w_2)) and writing the two conditions as four linear equations in the unknowns (\tau,\alpha,\beta) (or their “e‑” counterparts). Solving the system yields a single value for (\tau), which therefore determines a unique point on (L). The algebraic manipulation reduces to the observation that the determinant (w_1y_T-w_2x_T) equals (w_1y_S-w_2x_S) because both (S) and (T) lie on the same line (L), and that the expression (y_Sx_T-x_Sy_T+a_Sy_T) simplifies to (y_Sa_T) because (S) and (T) satisfy the equations of the parallel lines. Hence the two derived values of (\tau) coincide, establishing existence and uniqueness.

Proposition 2
The situation of Proposition 1 is embedded in a more general configuration: a fourth line called “Axis” (different from (L)) is introduced, together with a point (Origin) on Axis but not on (L). The same construction yields a unique point (P\in L) with analogous distance‑equality properties relative to the lines (Z_S) and (Z_T). By choosing Axis to be the x‑axis or y‑axis, one recovers (P_{\text{hor}}) and (P_{\text{ver}}) respectively, showing that Proposition 2 is equivalent to Proposition 1. Lemma 2 formalises this equivalence.

Proposition 3
Two parallel lines (G) and (P) are given, with (G) not passing through the origin. A fixed positive number (\varepsilon) is chosen. For an arbitrary point ((b_x,b_y)\in G) with (b_y\neq0), the points
\