We explore triangles with the property that the circumference of the incircle equals one of the side lengths. The main problem we solve is finding a way to characterise such triangles through their Cartesian coordinates. More precisely, given a triangle with a side between ±pi on the xaxis, and whose incentre has coordinates (u,1), what are the coordinates of the third point of the triangle? We also find the range of values of u for which this holds.

00:00 Intro
00:20 Existence & construction
00:56 Possible construction issue
01:28 Posing the problem
02:44 Equations for the sides
04:19 Gradient of L_1
07:05 Gradient of L_2
09:00 Intersection: xcoordinate
14:30 Intersection: ycoordinate
17:33 Range of values of u