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近日笔者翻看《给数学迷的500个挑战性问题》[1]一书,其中问题314:如图1,已知AB是圆O的直径,X是圆上除A与B外的一点,t_A,t_B,t_X分别是圆在A,B,X处的切线,设AX的延长线交t_B于Z,BX的延长线交t_A于Y。证明:YZ,t_X,AB三条直线或者交于一点,或者相互平行。从本题的题设可以发现,若直线tX交t_A于W,直线t_X交t_B于U,则W,U分别是线段AY,BZ的中点。证明:若YZ,t_X,AB三条直线相互平行,则W,U
Recently I read the “500 challenging mathematical problems to the math” [1] a book, in which the problem 314: As Figure 1, known AB is the diameter of the circle O, X is the circle except A and B points, t_A, t_B, t_X are the tangent of the circle at A, B and X, respectively. Let the extension of AX be t_B and the extension of ZB and BX be t_A at Y. Prove: YZ, t_X, AB three straight lines or at one point, or parallel to each other. From the title of this question can be found, if t_A straight line t_A in W, t_X intersect t_B in U, then W, U, respectively, the midpoint of the line segment AY, BZ. Proof: If YZ, t_X, AB three straight lines parallel to each other, then W, U