Problems in Geometry (1)
1. In a triangle ABC with orthocenter H and circumcircle O prove that
< BAH =< OAC .
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2. Let ABC be a triangle with orthocenter H in which AH, BH, CH meet BC, CA, AB
in D, E, F respectively. Prove that
HA · HD = HB · HE = HC · HF.
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3. Let ABC be a triangle with orthocenter H in which AH, BH, CH meet BC, CA, AB
in D, E, F respectively.
(A ) If ABC is an acute angled triangle, prove that H is the incenter of DEF.
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(B ) What happens if A is an obtuse angle ?
4. Let ABC be a triangle with incenter I and excenters Ia , Ib , Ic .
(A) Prove that I is the orthocenter of the triangle Ia Ib Ic .
(B) Prove that the circumcircle of a triangle passes through the midpoint of a line
segment joining any two of the excenters.
(C) Prove that the circumcircle of a triangle passes through the midpoint of a line
segment joining the incenter with any one of the excenters.
5. In a triangle ABC with centroid G, let AG, BG, CG meet BC, CA, AB in A0 , B 0 , C 0
respectively. Let the parallel to AB through C intersect B 0 C 0 in K. Prove that
(A) AK is parallel to CC 0 and |AK| = |CC 0 | , |A0 K| = |BB 0 | .
(B) Prove that the the line CA is the median of the triangle AA0 K through the point
A.
(C) Prove that the length of the median of the triangle AA0 K through the point A is
3b/4.
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Consider the right triangle BDA where D is the foot of the altitude through A and the isosceles
triangle OCA.
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Note that the triangles HDC and HF A are similar.
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Note that D, H, E, C are concyclic and hence < EDH =< ECH = 90 − A .
6. In a triangle ABC, let ma , mb , mc denote the lengths of the medians through A, B, C
respectively.
(A) Prove that 2ma ≤ b + c .
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(B) Prove that ma + mb + mc ≤ a + b + c .
(C) Prove that
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a + b + c ≤ ma + mb + mc .
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Can any of these inequalities degenerate into an equality ? 5
7. Let ABC be a triangle with orthocenter H and centroid G. Let Ga , Gb , Gc be the
centroids of HBC, HCA, HAB respectively.
(A) Prove that Gb Gc is parallel to BC.
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(B) Prove that G is the orthocenter of the triangle Ga Gb Gc .
8. In a triangle ABC, let k, l, m be the perpendicular bisectors of [B, C], [C, A], [A, B].
Given Xn for each n ∈ Z with X3n ∈ k, X3n+1 ∈ l, X3n+2 ∈ m such that A ∈ X3n+1 X3n+2 ,
B ∈ X3n+2 X3n+3 , C ∈ X3n X3n+1 , prove that Xn = Xn+6 for all n ∈ Z. 7
9. Consider a triangle ABC with incenter I. Given Xn for each n ∈ Z with X3n ∈
BC, X3n+1 ∈ CA, X3n+2 ∈ AB such that IC ⊥ X3n X3n+1 , IA ⊥ X3n+1 X3n+2 , IB ⊥
X3n+2 X3n+3 for all n ∈ Z, prove that Xn = Xn+6 for all n ∈ Z.
Let A0 be the midpoint of [B, C]. Let the point T be chosen on AA0 such that A0 is also the midpoint
of [A, T ]. Consider the paralelogram ABT C and the triangle AT C.
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Remember the triangle AA0 K in Problem 5. Its sidelengths are ma , mb , mc . What are the lengths
of the medians of AA0 K? .
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Remember that in the triangle HBC the point Gb divides [HA0 ] in the ratio −1 : 2 . Similar
observations in ABC, HCA, HAB .
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Start with X0 ∈ k and construct X1 ∈ l, X2 ∈ m and so on. Let α =< X0 CB . Compute the values
of < X1 AC , < X2 BA , < X3 CB , < X4 AC , < X5 BA . What do you notice ?
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