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150 Nice Geometry Problems - Amir Hossein Parvardi.
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150 Nice Geometry Problems - Amir Hossein Parvardi.
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Geometry ProblemsAmir Hossein Parvardi∗
January 9, 2011
Edited by: Sayan Mukherjee.
Note. Most of problems have solutions. Just click on the number beside
the problem to open its page and see the solution! Problems posted by different
authors, but all of them are nice! Happy Problem Solving!
1. Circles W1 , W2 intersect at P, K. XY is common tangent of two circles
which is nearer to P and X is on W1 and Y is on W2 . XP intersects W2 for the
second time in C and Y P intersects W1 in B. Let A be intersection point of
BX and CY . Prove that if Q is the second intersection point of circumcircles
of ABC and AXY
∠QXA = ∠QKP
2. Let M be an arbitrary point on side BC of triangle ABC. W is a circle
which is tangent to AB and BM at T and K and is tangent to circumcircle
of AM C at P . Prove that if T K||AM , circumcircles of AP T and KP C are
tangent together.
3. Let ABC an isosceles triangle and BC > AB = AC. D, M are respec-
tively midpoints of BC, AB. X is a point such that BX ⊥ AC and XD||AB.
BX and AD meet at H. If P is intersection point of DX and circumcircle of
AHX (other than X), prove that tangent from A to circumcircle of triangle
AM P is parallel to BC.
4. Let O, H be the circumcenter and the orthogonal center of triangle
4ABC, respectively. Let M and N be the midpoints of BH and CH. Define
∗ Email: ahpwsog@gmail.com, blog: http://www.math- olympiad.blogsky.com/
1
,B 0 on the circumcenter of 4ABC, such that B and B 0 are diametrically opposed.
1
If HON M is a cyclic quadrilateral, prove that B 0 N = AC.
2
5. OX, OY are perpendicular. Assume that on OX we have wo fixed points
P, P 0 on the same side of O. I is a variable point that IP = IP 0 . P I, P 0 I
intersect OY at A, A0 .
a) If C, C 0 Prove that I, A, A0 , M are on a circle which is tangent to a fixed
line and is tangent to a fixed circle.
b) Prove that IM passes through a fixed point.
6. Let A, B, C, Q be fixed points on plane. M, N, P are intersection points
of AQ, BQ, CQ with BC, CA, AB. D0 , E 0 , F 0 are tangency points of incircle of
ABC with BC, CA, AB. Tangents drawn from M, N, P (not triangle sides) to
incircle of ABC make triangle DEF . Prove that DD0 , EE 0 , F F 0 intersect at Q.
7. Let ABC be a triangle. Wa is a circle with center on BC passing through
A and perpendicular to circumcircle of ABC. Wb , Wc are defined similarly.
Prove that center of Wa , Wb , Wc are collinear.
8. In tetrahedron ABCD, radius four circumcircles of four faces are equal.
Prove that AB = CD, AC = BD and AD = BC.
9. Suppose that M is an arbitrary point on side BC of triangle ABC. B1 , C1
are points on AB, AC such that M B = M B1 and M C = M C1 . Suppose that
H, I are orthocenter of triangle ABC and incenter of triangle M B1 C1 . Prove
that A, B1 , H, I, C1 lie on a circle.
10. Incircle of triangle ABC touches AB, AC at P, Q. BI, CI intersect with
P Q at K, L. Prove that circumcircle of ILK is tangent to incircle of ABC if
and only if AB + AC = 3BC.
2
, 11. Let M and N be two points inside triangle ABC such that
∠M AB = ∠N AC and ∠M BA = ∠N BC.
Prove that
AM · AN BM · BN CM · CN
+ + = 1.
AB · AC BA · BC CA · CB
12. Let ABCD be an arbitrary quadrilateral. The bisectors of external
angles A and C of the quadrilateral intersect at P ; the bisectors of external
angles B and D intersect at Q. The lines AB and CD intersect at E, and the
lines BC and DA intersect at F . Now we have two new angles: E (this is the
angle ∠AED) and F (this is the angle ∠BF A). We also consider a point R of
intersection of the external bisectors of these angles. Prove that the points P ,
Q and R are collinear.
13. Let ABC be a triangle. Squares ABc Ba C, CAb Ac B and BCa Cb A
are outside the triangle. Square Bc Bc0 Ba0 Ba with center P is outside square
ABc Ba C. Prove that BP, Ca Ba and Ac Bc are concurrent.
14. Triangle ABC is isosceles (AB = AC). From A, we draw a line ` parallel
to BC. P, Q are on perpendicular bisectors of AB, AC such that P Q ⊥ BC.
π
M, N are points on ` such that angles ∠AP M and ∠AQN are . Prove that
2
1 1 2
+ ≤
AM AN AB
15. In triangle ABC, M is midpoint of AC, and D is a point on BC such
that DB = DM . We know that 2BC 2 − AC 2 = AB.AC. Prove that
AC 2 .AB
BD.DC =
2(AB + AC)
16. H, I, O, N are orthogonal center, incenter, circumcenter, and Nagelian
point of triangle ABC. Ia , Ib , Ic are excenters of ABC corresponding vertices
A, B, C. S is point that O is midpoint of HS. Prove that centroid of triangles
Ia Ib Ic and SIN concide.
17. ABCD is a convex quadrilateral. We draw its diagonals to divide the
quadrilateral to four triangles. P is the intersection of diagonals. I1 , I2 , I3 , I4 are
3
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