Posts tagged 'hyperbolic-functions'

F.19 Equal Incircles Theorem

If a fan of rays from apex P to a baseline cuts triangle ABC into consecutive subtriangles whose incircles have equal radii r_1 = r_2 = … = r_n = r, then the ray angles gamma_n with the altitude PR (of height h) follow the hyperbolic progression tan(gamma_n) = sinh(a + n*b) with tanh(b/2) = r/(h - r), so overlapping pairs of subtriangles also have equal incircles.