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Very Special Triangles

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 x-axis, 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: x-coordinate 14:30 Intersection: y-coordinate 17:33 Range of values of u