We consider two connected problems:
1. Constructing a triangle which meets another
These problems beg the question of what is meant by ‘meet’ and we now aim to make this precise:
Definition: A triangle XYZ will meet a given triangle ABC if on the triangle ABC, the vertex X lies on a line through AB, the vertex Y lies on a line through BC, and the vertex Z lies on a line through CA.
When triangle XYZ is actually ‘in’ the triangle ABC, ‘meet’ is synonymous with the traditional ‘inscribe’ (such as in case (1) below). For ‘inscribe’ we understand that some of X, Y, Z may coincide with the vertices of ABC (such as case (2) below).
More generally we use ‘meet’ to extend these possibilities by allowing XYZ to meet triangle ABC with its sides produced externally (such as case (3) below).