Forum Geometricorum Volume 16 (2016) 85–94. FORUM GEOM ISSN 1534-1178 Solutions of Two Japanese Ellipse Problems J. Marshall Unger Abstract. Two premodern Japanese theorems involving an ellipse and tangent circles are stated as problems in [1] (6.4.7 and 6.2.4). Neither can be proven easily by elementary means. But a proof of the second problem follows from another Japanese proposition, for which I give an original proof. It is based on the first part of an elegant proof of a generalization of the first theorem, which I present in edited form. 1. Introduction Among the many theorems involving ellipses stated as problems in [1], two (6.4.7 and 6.2.4) stand out as particularly challenging. The first theorem (Figure 1) concerns two intersecting tangents to an ellipse and the circles that touch both tangents and the ellipse. If the diameters of the two that touch the ellipse externally are d 1 , d 4 , and those of the two that touch it internally are d 2 , d 3 , then d 1 : d 2 = d 3 : d 4 . d 1 2 d 2 2 d 3 2 d 4 2 Figure 1 The second theorem states that, in Figure 2, v = (R − r) √ Rr + u 2 − u(R + r) 2 √ Rr , where u, v are the semi-major and semi-minor axes of the ellipse and R, r are the radii of the two circles touching the ellipse and the sides of the square that are touched by it. Publication Date: March 29, 2016. Communicating Editor: Paul Yiu.