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Geometry_Problem_Set_I (1)
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Geometry Problem Set IMaths Olympiad Preparation
24th January 2024
1 Problems
1. Let G be the centroid of ∆ABC such that AG, BG & CG extended
meet the circumcircle at points A′ , B ′ & C ′ respectively. Prove
that
AG BG CG
+ + = 3.
GA′ GB ′ GC ′
2. Let D & E b two isogonal points on the side BC of ∆ABC such
that AD & AE extended meet the circumcircle at points X & Y
respectively. Prove that
AD AE
= .
DX EY
3. Let G be the centroid of ∆ABC and AG extended meets the side
BC at point D and the circumcircle at point A′ . Define the points
E, B ′ , F & C ′ analogously. Prove that
A′ D B′E C ′F 1
′
+ ′
+ ′
= .
3A D + AD 3B E + BE 3C F + CF 2
4. Let O be the circumcentre of ∆ABC such that AO extended
meets the side BC at point P and the circumcircle at point A′ .
Define the points Q, B ′ , R & C ′ respectively. Prove that
A′ P B′Q C ′R
+ + = 1.
AP BQ CR
1
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5. Let O, N and R be the circumcentre, nine-point centre and cir-
cumradius of ∆ABC respectively. Let X, Y & Z be the feet of
the perpendiculars from N upon sides BC, CA & AB respectively.
Prove that
3abcR
a3 · N X + b3 · N Y + c3 · N Z = .
2
6. Let G & r be the centroid & inradius of ∆ABC such that D, E
& F are the feet of the perpendiculars from G onto the sides
BC, CA & AB respectively. Prove that
1 1 1 3
+ + = .
GD GE GF r
7. In ∆ABC, prove that
∠A = 4∠B ⇐⇒ bc2 (b + c) = (a2 − b2 )(a2 − b2 − bc).
8. Let H be the orthocentre of ∆ABC such that AH, BH & CH
extended meet their opposite sides at D, E & F respectively.
Let Ha , Hb & Hc be the orthocentres of ∆AEF, ∆BF D & ∆CDE
respectively. Prove that the line segments DHa , EHb & F Hc are
concurrent & each of them is bisected at the point of concurrency
and
∆Ha Hb Hc ∼
= ∆DEF.
2 Maths Olympiad Preparation
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