Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.
Click the figure below to see the complete problem 627.
Wednesday, June 22, 2011
Problem 627: Cyclic Quadrilateral, Congruence, Concyclic Points
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Denote each of the equal angles ABD, ACD by x
ReplyDeleteAB = BE, CD=CF imply that
each of the angles AEB, CFD is (180° - x)/2
Taking supplements,
angle AED = angle AFD
So A,E,F,D are concyclic
The internal bisectors of the angles ABD and ACD
ReplyDeletemeet at the midpoint O of the (minor)arc AD.
Also O lies on the perpendicular bisector of AE as well as FD.
So OE = OA = OD = OF = k say
∴A, E, F, D lie on the circle (O;k)
Since < ABE = < ACD, Tr.s ABE & FCD are similar, hence < AED = < AFD and the result follows.
ReplyDeleteSumith Peiris
Moratuwa
Sri Lanka