[Paper Review] Studies in Similarity

Christopher Bradley|arXiv (Cornell University)|Jul 7, 2010

Mathematics and Applications3 references3 citations

TL;DR

This paper investigates nine circles derived from a triangle's vertices, orthocenter, and Brocard points, showing their centers form six triangles each similar to the original triangle—three directly similar and three indirectly similar. The three indirectly similar triangles are mutually in perspective with the circumcenter as the perspector, revealing a deep geometric symmetry tied to the orthocentroidal circle and areal coordinates.

ABSTRACT

Three circles define each of the Brocard points of a triangle. If one adds the three circles through a pair of vertices and the orthocentre one has nine circles. It is described how each of the nine centres of these circles lies at the vertex of two triangles producing six triangles. These triangles are each similar to the original triangle, three being directly similar and three indirectly similar. These latter three are mutually in perspective with the circumcentre as perspector.

Motivation & Objective

• To identify and characterize nine circles formed by pairs of vertices and key triangle centers—orthocenter, Brocard points, and three new points aH, bH, cH.
• To determine the geometric and algebraic properties of the nine circle centers and their configuration in the plane.
• To prove that these nine centers form six triangles, each similar to the original triangle ABC, with three direct and three indirect similarities.
• To establish that the three indirectly similar triangles share the circumcenter O as their common perspector.
• To integrate these findings into the broader framework of direct and indirect similarities in triangle geometry, particularly in relation to Hagge circles and Wood’s classification.

Proposed method

• Using areal coordinates, the paper derives the equations of nine circles: three through the orthocenter H (BHC, CHA, AHB), three through the Brocard point H+ (AH+B, BH+C, CH+A), and three through H− (AH−B, BH−C, CH−A).
• The centers of these nine circles are computed using the pole of the line at infinity, yielding unnormalized areal coordinates for each center.
• The points aH, bH, cH are identified as the second intersections of the medians with the orthocentroidal circle (diameter GH), and their areal coordinates are derived algebraically.
• The paper applies the areal metric to compute squared distances between centers, showing proportionality to the squares of the original triangle’s side lengths, thus proving similarity.
• It verifies that triangles formed from the nine centers (e.g., cA aB bC, aA aB aC) are similar to ABC by showing (side1)² ∝ a², (side2)² ∝ b², (side3)² ∝ c².
• It demonstrates that the three triangles formed from the aA, bB, cC centers are mutually in perspective with the circumcenter O, using collinearity and concurrency arguments.

Experimental results

Research questions

• RQ1 How do the nine centers of the circles through pairs of vertices and the orthocenter, Brocard points, and the three new points aH, bH, cH relate geometrically?
• RQ2 What is the nature of the six triangles formed by these nine centers, and how do they relate in similarity to the original triangle ABC?
• RQ3 Why are the three triangles formed from the aA, bB, cC centers indirectly similar to ABC and mutually in perspective with the circumcenter O?
• RQ4 What is the role of the orthocentroidal circle in locating the points aH, bH, cH and in unifying the nine-circle configuration?
• RQ5 How do these configurations fit into the broader theory of direct and indirect similarities in triangle geometry, particularly in relation to Hagge circles and Wood’s framework?

Key findings

• The nine centers of the circles through pairs of vertices and H, H+, H− lie at the vertices of six triangles, three directly similar and three indirectly similar to triangle ABC.
• The three indirectly similar triangles are mutually in perspective with the circumcenter O as the perspector, a result confirmed via collinearity of corresponding lines from the centers to O.
• The points aH, bH, cH lie on the orthocentroidal circle (diameter GH), and their areal coordinates are aH(a², b² + c² – a², b² + c² – a²), bH(c² + a² – b², b², c² + a² – b²), cH(a² + b² – c², a² + b² – c², c²).
• The squared distances between centers are proportional to the squares of the original triangle’s side lengths, confirming similarity: (bC aB)² = k a², (cA bC)² = k b², (aB cA)² = k c² for some scalar k.
• The triangles aA aB aC, bA bB bC, and cA cB cC are all directly similar to ABC and to each other, with area ratios l : m : n, where l, m, n are derived from side-length expressions.
• The three triangles formed from centers aA, bB, cC are mutually in perspective with O, and their similarity is confirmed via areal metric computations showing (aB aC)² ∝ a², (aC aA)² ∝ b², (aA aB)² ∝ c².