Proposed Problem
Click the figure below to see the complete problem 301 about Tangents to a circle, Secants, Major Arc.
See also:
Complete Problem 301
Collection of Geometry Problems
Level: High School, SAT Prep, College geometry
Saturday, June 13, 2009
Problem 301: Tangents to a circle, Secants, Major Arc
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Join OA and draw FF' and EE' perpendicular to OA meeting AC & AB bothe extended in F' & E'. Complete the isosceles trapezium EE'F'F which also is concyclic. We can easily see that CF'=BF=a. Hence, EF'=EC+CF'=b+a. Also CE=BE' since BCEE' is also an isosceles trapezium. Further, triangles BEE' & FCF' are similar; hence, FF'/a = b/EE' or EE'*FF'=ab. Now apply Ptolemy's Theorem to concyclic E'EFF' to get E'E*F'F + EF*E'F' = EF'*E'F or ab + x^2 =(a+b)^2 which gives, x^2 = a^2 + b^2 + ab. QED.
ReplyDelete--- Delightful problems these two - 300 & 301 with very neat results. Trust my solutions are found alright.
Ajit: ajitathle@gmail.com