# A Chengdu examination math problem analysis, comprehensive strong, difficult

As shown in the figure, take a point O on the edge of BC of triangle ABC and draw a circle with point O as the center and OC as the radius.Circle O is tangent to edge AB at point D, AC=AD, connects OA intersecting circle O at point E, connects CE, and extends intersection segment at point F.(2) If AB=10, tanB=4/3, find the radius of circle O (3) if F is the midpoint of AB, try to explore the quantitative relationship between BD+CE and AF, and explain the reason.(1) To prove that AC is tangent, we can prove that AC perpendicular OC.By observing the graph and connecting OD with AD=AC, △ACO≌△ADO (the three sides are equal) is obviously shown.B b b b b b b B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B BBecause BG/BO=DB/AB, which is x/5x=3x/10, x=2/3 so the radius is 4x=8/3.ED=EC, ∠CEO=∠DEO, OC=OE=OD∠OCE=∠CEO=∠ ODE, so ∠DEF=180°-2∠ECO.In the right triangle ABC, F is the midpoint of the hypotenuse, so CF=AF=BF, so ∠FCB=FBC, in the triangle FBC, Angle CFB=180°-2∠FCB = Angle DEF= DFE so DE=DF=EC, AF=BF=CE+BD.Summary: this topic is more comprehensive or strong, in the exam is a more difficult last question.The core of this paper is similar triangles and finding the maximum value of functions