Infographic Geometry Problem 1056: Triangle, Exradius, Reciprocals of the Altitudes, Multiplicative Inverse, Perpendicular, Excircle, Circle, Tangent. Level: High School, College, Mathematics Education

In a triangle ABC with the altitudes h_a, h_b, and h_c and the exradii r_a, r_b, r_c, prove that: 1/r_a = 1/h_b + 1/h_c - 1/h_a, similarly, 1/r_b = 1/h_a + 1/h_c - 1/h_b, and 1/r_c = 1/h_a + 1/h_b - 1/h_c. (See the figure below)