The tie theorems

TL;DR

This paper presents a new generalization of the tie theorems in projective geometry, establishing a common intersection point for certain lines and exploring related properties.

Contribution

It introduces a generalized version of the tie theorems in projective geometry and uncovers new related geometric properties.

Findings

Lines CW_C, AW_A, and BW_B are concurrent.

Generalization of the tie theorems to broader configurations.

Discovery of additional geometric properties related to the theorem.

Abstract

Theorem. There are general position points A, B, C, P on the projective plane. Let A_P be the intersection point of lines AP and BC. Analogously define B_P and C_P. Take any points A_1, B_1, C_1 on AP, BP, CP, respectively. Let W_C be the intersection point of A_1B_P and B_1A_P. Analogously define points W_A and W_B. Then lines CW_C, AW_A and BW_B pass through one point. We also generalize this theorem and find interesting related properties.