MathDB

5

Part of 2011 India Regional Mathematical Olympiad

Problems(2)

Indian RMO 2011: Question 5

Source:

12/4/2011
Let ABCABC be a triangle and let BB1,CC1BB_1,CC_1 be respectively the bisectors of ∠B,∠C\angle{B},\angle{C} with B1B_1 on ACAC and C1C_1 on ABAB, Let E,FE,F be the feet of perpendiculars drawn from AA onto BB1,CC1BB_1,CC_1 respectively. Suppose DD is the point at which the incircle of ABCABC touches ABAB. Prove that AD=EFAD=EF
geometryincentertrigonometrygeometric transformationreflectionAMCAIME
Convex quadrilateral and midpoints [RMO2-2011, India]

Source:

12/31/2011
Let ABCDABCD be a convex quadrilateral. Let E,F,G,HE,F,G,H be the midpoints of AB,BC,CD,DAAB,BC,CD,DA respectively. If AC,BD,EG,FHAC,BD,EG,FH concur at a point O,O, prove that ABCDABCD is a parallelogram.
geometryanalytic geometryprojective geometry