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Geometry Problem 1163: Isosceles Triangle, Congruence, Double Angle, Circle, Concyclic Points, Circumcenter, Perpendicular, Angle Bisector. Level: School, College, Mathematics Education

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The figure below shows a point D inside an isosceles triangle ABC (AB = BC) so that angle DCB = 2 angle BAD and DC = BC. BD extended meets AC at G, CD extended meets AB at H, and the bisector of angle ABC meets AD extended at N. Prove that (1) GN is perpendicular to BC at M; (2) ABNG is concyclic at O1; (3) ACNH is concyclic at O2; (4) CGDN is concyclic at O3; (5) BCGH is concyclic at O4; (6) O1NO4O3O2 is concyclic at G; (7) G is the circumcenter of triangle ADH.
This entry contributed by Sumith Peiris, Moratuwa, Sri Lanka.

 

Geometry Problem 1163 Isosceles Triangle, Congruence, Double Angle, Circle, Concyclic Points, Circumcenter, Perpendicular, Angle Bisector

 


See also: Geometry art of problem 1163

 

 

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Home | SearchGeometry | Problems | All Problems | Open Problems | Visual Index | 10 Problems | Problems Art Gallery | Art | 1161-1170 | Triangles | Circle | Circumcenter | Congruence | Triangle with double angle | Perpendicular lines | Congruence | Cyclic Quadrilateral | Concyclic Points | Angle Bisector | Last updated: Nov 7, 2015

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