Posts tagged 'sangaku'

F.9 Circle Equations

Pairwise subtraction of Cartesian circle equations (x - u_i)^2 + (y - v_i)^2 = r_i^2 eliminates the quadratic terms x^2 + y^2 to yield linear radical-axis relations, determining both the side Z of a square from three corner distances and the radius of a Sangaku circle tangent to an inscribed square and equilateral triangle.

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.