Some Problems on Geometry
December 16, 2019
1. Suppose ABCDE is a convex pentagon satisfying ∠ABC = 120◦ , ∠CDE = 60◦ ,
AB = BC, CD = DE, and BD = 2. Find the area of ABCDE.
√
√
√
2. For a, b, c > 0, prove the inequality a2 − ab + b2 + b2 − bc + c2 ≥ a2 + ac + c2 .
Furthermore, show that equality holds if and only if 1/b = 1/a + 1/c.
3. Let ABC be a triangle with AB = 13, BC = 14, CA = 15. Let O be its circumcentre and let AD ⊥ BC with D on BC. Suppose that X is on DC and Y is on
AD such that XY k AO and AX ⊥ Y O. Find the length of BX.
4. Suppose ABCD is a cyclic quadrilateral and r1 , r2 , r3 , r4 are respectively the inradii
of triangles ABC, ADC, BAD, BCD respectively. Show that r1 + r2 = r3 + r4 .
5. Let ABCD be a cyclic quadrilateral inscribed in a circle Γ. Let E, F, G, H be the
midpoints of arcs AB, BC, CD, AD of Γ, respectively. Suppose that AC · BD =
EG · F H. Show that AC, BD, EG, F H are all concurrent.
6. Let ABCD be a convex quadrilateral. Let diagonals AC and BD intersect at P .
Let P E, P F, P G and P H are altitudes from P on the side AB, BC, CD and DA
respectively. Show that ABCD has a incircle if and only if
1
1
1
1
+
=
+
.
PE PG
PF
PH
7. In an acute triangle ABC, let O, G, H be its circumcentre, centroid and orthocentre, respectively. Let D ∈ BC, E ∈ CA such that OD ⊥ BC, HE ⊥ CA. Let F be
the midpoint of AB. If the triangles ODC, HEA, GF B have the same area, find
all the possible values of ∠C.
8. Let ABC be a right-angle triangle with ∠B = 90◦ . Let D be a point on AC such
that the inradii of the triangles ABD and CBD are equal. If this common value
is r0 and if r is the inradius of triangle ABC, prove that 1/r0 = 1/r + 1/BD.
9. Let ABC be an acute angled triangle. Let D, E, F be points on BC, CA, AB
such that AD is the median, BE is the internal bisector and CF is the altitude.
Suppose that ∠F DE = ∠C, ∠DEF = ∠A and ∠EF D = ∠B. Show that ABC is
equilateral.
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10. Let ABC be a triangle with AB = AC. The angle bisectors of ∠CAB and ∠ABC
meet the sides BC and CA at D and E, respectively. Let K be the incentre of
triangle ADC. Suppose that ∠BEK = 45◦ . Find all possible values of ∠CAB.
11. Given an acute triangle ABC with O as its circumcentre. Line AO intersects BC
at D. Points E, F are on AB, AC respectively such that A, E, D, F are concyclic.
Prove that the length of the projection of line segment EF on side BC does not
depend on the positions of E and F .
12. Consider an acute-angled triangle ABC. Let P be the foot of the altitude of
triangle ABC issuing from the vertex A, and let O be the circumcenter of triangle
ABC. Assume that ∠C ≥ ∠B + 30◦ . Prove that ∠A + ∠COP < 90◦ .
13. Let ABC be an acute triangle. Let ω be a circle whose centre L lies on the side
BC. Suppose that ω is tangent to AB at B 0 and AC at C 0 . Suppose also that the
circumcenter O of triangle ABC lies on the shorter arc B 0 C 0 of ω. Prove that the
circumcircle of ABC and ω meet at two points.
14. Let O be the circumcenter and H the orthocenter of an acute triangle ABC. Show
that there exist points D, E, and F on sides BC, CA, and AB respectively such
that OD + DH = OE + EH = OF + F H and the lines AD, BE, and CF are
concurrent.
15. Let H be the orthocenter of an acute-angled triangle ABC. The circle ΓA centered
at the midpoint of BC and passing through H intersects the sideline BC at points
A1 and A2 . Similarly, define the points B1 , B2 , C1 and C2 . Prove that the six
points A1 , A2 , B1 , B2 , C1 and C2 are concyclic.
16. Suppose that a > b > c > d are positive integers such that
ac + bd = (b + d + a − c)(b + d − a + c).
Prove that ab + cd can not be a prime.
email: ghoshadi26@gmail.com
Sources of the above problems:
1. RMO 2008,
2. Unknown, 3. inspired from HMMT 2014, 4. Unknown,
5. INMO 2011,
6. INMO 2015, 7. INMO 2013, 8. INMO 2016,
9. INMO 2012,
10. IMO 2009,
11. CGMO 2004, 12. IMO 2001,
13. IMOSL 2011, 14. IMOSL 2000, 15. IMO 2008,
16. IMO 2001.
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