Subcontests
(64)Geometry Mathley 16.3 concurrent
The incircle (I) of a triangle ABC touches BC,CA,AB at D,E,F. Let ID,IE,IF intersect EF,FD,DE at X,Y,Z, respectively. The lines ℓa,ℓb,ℓc through A,B,C respectively and are perpendicular to YZ,ZX,XY .
Prove that ℓa,ℓb,ℓc are concurrent at a point that is on the line segment joining I and the centroid of triangle ABC .Nguyễn Minh Hà Geometry Mathley 16.2 collinear orthocenters
Let ABCD be a quadrilateral and P a point in the plane of the quadrilateral. Let M,N be on the sides AC,BD respectively such that PM∥BC,PN∥AD. AC meets BD at E. Prove that the orthocenter of triangles EBC,EAD,EMN are collinear if and only if P is on the line AB.Đỗ Thanh SơnPS. Instead of the word collinear, it was written concurrent, probably a typo. Geometry Mathley 16.1 four concurrent Euler lines
Let ABCD be a cyclic quadrilateral with two diagonals intersect at E. Let M, N, P, Q be the reflections of E in midpoints of AB, BC, CD, DA respectively. Prove that the Euler lines of △MAB, △NBC, △PCD, △QDA are concurrent.Trần Quang Hùng Geometry Mathley 15.3 OH^2/ R^2=OM/ OD +ON /OE + OP / OF
Triangle ABC has circumcircle (O,R), and orthocenter H. The symmedians through A,B,C meet the perpendicular bisectors of BC,CA,AB at D,E,F respectively. Let M,N,P be the perpendicular projections of H on the line OD,OE,OF. Prove that R2OH2=ODOM+OEON+OFOP
Đỗ Thanh Sơn Geometry Mathley 15.2 concyclic
Let O be the centre of the circumcircle of triangle ABC. Point D is on the side BC. Let (K) be the circumcircle of ABD. (K) meets AO at E that is distinct from A.
(a) Prove that B,K,O,E are on the same circle that is called (L).
(b) (L) intersects AB at F distinct B. Point G is on (L) such that EG∥OF. GK meets AD at S,SO meets BC at T . Prove that O,E,T,C are on the same circle.Trần Quang Hùng Geometry Mathley 15.1 concurrent circumcircles, incircle related
Let ABC be a non-isosceles triangle. The incircle (I) of the triangle touches sides BC,CA,AB at A0,B0, and C0. Points A1,B1, and C1 are on BC,CA,AB such that BA1=CA0,CB1=AB0,AC1=BC0. Prove that the circumcircles (IAA1),(IBB1),(ICC1) pass all through a common point, distinct from I.Nguyễn Minh Hà Geometry Mathley 14.4 concurrency related to NPC centers
Two triangles ABC and PQR have the same circumcircles. Let Ea,Eb,Ec be the centers of the Euler circles of triangles PBC,QCA,RAB. Assume that da is a line through Ea parallel to AP, db,dc are defined in the same manner. Prove that three lines da,db,dc are concurrent.Nguyễn Tiến Lâm, Trần Quang Hùng Geometry Mathley 14.3 <DGE + <FGQ = 180^o
Let ABC be a triangle inscribed in circle (I) that is tangent to the sides BC,CA,AB at points D,E,F respectively. Assume that L is the intersection of BE and CF,G is the centroid of triangle DEF,K is the symmetric point of L about G. If DK meets EF at P,Q is on EF such that QF=PE, prove that ∠DGE+∠FGQ=180o.Nguyễn Minh Hà Geometry Mathley 14.2 angle bisector related to Euler Circle
The nine-point Euler circle of triangle ABC is tangent to the excircles in the angle A,B,C at Fa,Fb,Fc respectively. Prove that AFa bisects the angle ∠CAB if and only if AFa bisects the angle ∠FbAFc.Đỗ Thanh Sơn
Geometry Mathley 14.1 equal circles
A circle (K) is through the vertices B,C of the triangle ABC and intersects its sides CA,AB respectively at E,F distinct from C,B. Line segment BE meets CF at G. Let M,N be the symmetric points of A about F,E respectively. Let P,Q be the reflections of C,B about AG. Prove that the circumcircles of triangles BPM,CQN have radii of the same length.Trần Quang Hùng
Geometry Mathley 13.4 concurrent circles, similar triangles
Let P be an arbitrary point in the plane of triangle ABC. Lines PA,PB,PC meets the perpendicular bisectors of BC,CA,AB at Oa,Ob,Oc respectively. Let (Oa) be the circle with center Oa passing through two points B,C, two circles (Ob),(Oc) are defined in the same manner. Two circles (Ob),(Oc) meets at A1, distinct from A. Points B1,C1 are defined in the same manner. Let Q be an arbitrary point in the plane of ABC and QB,QC meets (Oc) and (Ob) at A2,A3 distinct from B,C. Similarly, we have points B2,B3,C2,C3. Let (Ka),(Kb),(Kc) be the circumcircles of triangles A1A2A3,B1B2B3,C1C2C3. Prove that
(a) three circles (Ka),(Kb),(Kc) have a common point.
(b) two triangles KaKbKc,ABC are similar.Trần Quang Hùng Geometry Mathley 13.3 concurrent iff concyclic
Let ABCD be a quadrilateral inscribed in circle (O). Let M,N be the midpoints of AD,BC. A line through the intersection P of the two diagonals AC,BD meets AD,BC at S,T respectively. Let BS meet AT at Q. Prove that three lines AD,BC,PQ are concurrent if and only if M,S,T,N are on the same circle.Đỗ Thanh Sơn Geometry Mathley 13.2 Euler line of ABC is Newton line of PQRS
In a triangle ABC, the nine-point circle (N) is tangent to the incircle (I) and three excircles (Ia),(Ib),(Ic) at the Feuerbach points F,Fa,Fb,Fc. Tangents of (N) at F,Fa,Fb,Fc bound a quadrangle PQRS. Show that the Euler line of ABC is a Newton line of PQRS.Luis González
Geometry Mathley 12.4 orthogonal circles pairwise
QuadrilateralABCD has two diagonals AC,BD that are mutually perpendicular. Let M be the Miquel point of the complete quadrilateral formed by lines AB,BC,CD,DA. Suppose that L is the intersection of two circles (MAC) and (MBD). Prove that the circumcenters of triangles LAB,LBC,LCD,LDA are on the same circle called ω and that three circles (MAC),(MBD),ω are pairwise orthogonal.Nguyễn Văn Linh
Geometry Mathley 12.3 concurency
Points E,F are chosen on the sides CA,AB of triangle ABC. Let (K) be the circumcircle of triangle AEF. The tangents at E,F of (K) intersect at T . Prove that
(a) T is on BC if and only if BE meets CF at a point on the circle (K),
(b) EF,PQ,BC are concurrent given that BE meets FT at M,CF meets ET at N,AM and AN intersects (K) at P,Q distinct from A.Trần Quang Hùng Geometry Mathley 12.2 concurrency, fixed line
Let K be the midpoint of a fixed line segment AB, two circles (O) and (O′) with variable radius each such that the straight line OO′ is throughK and K is inside (O), the two circles meet at A and C, center O′ is on the circumference of (O) and O is interior to (O′). Assume that M is the midpoint of AC,H the projection of C onto the perpendicular bisector of segment AB. Let I be a variable point on the arc AC of circle (O′) that is inside (O),I is not on the line OO′ . Let J be the reflection of I about O. The tangent of (O′) at I meets AC at N. Circle (O′JN) meets IJ at P, distinct from J, circle (OMP) intersects MI at Q distinct from M. Prove that
(a) the intersection of PQ and O′I is on the circumference of (O).
(b) there exist a location of I such that the line segment AI meets (O) at R and the straight line BI meets (O′) at S, then the lines AS and KR meets at a point on the circumference of (O).
(c) the intersection G of lines KC and HB moves on a fixed line.Lê Phúc Lữ Geometry Mathley 12.1 fixed ratio HQ/HD
Let ABC be an acute triangle with orthocenter H, and P a point interior to the triangle. Points D,E,F are the reflections of P about BC,CA,AB. If Q is the intersection of HD and EF, prove that the ratio HQ/HD is independent of the choice of P.Luis González Geometry Mathley 11.4 6 concurrent circles
Let ABC be a triangle and P be a point in the plane of the triangle. The lines AP,BP,CP meets BC,CA,AB at A1,B1,C1, respectively. Let A2,B2,C2 be the Miquel point of the complete quadrilaterals AB1PC1BC, BC1PA1CA, CA1PB1AB. Prove that the circumcircles of the triangles APA2,BPB2, CPC2, BA2C, AB2C, AC2B have a point of concurrency.Nguyễn Văn Linh Geometry Mathley 11.2 AP = AQ , BM = BC = CN
Let ABC be a triangle inscribed in the circle (O). Tangents at B,C of the circles (O) meet at T . Let M,N be the points on the rays BT,CT respectively such that BM=BC=CN. The line through M and N intersects CA,AB at E,F respectively; BE meets CT at P,CF intersects BT at Q. Prove that AP=AQ.Trần Quang Hùng Geometry Mathley 11.1 hexagon inequality
Let ABCDEF be a hexagon with sides AB,CD,EF being equal to m units, sides BC,DE,FA being equal to n units. The diagonals AD,BE,CF have lengths x,y, and z units. Prove the inequality xy1+yz1+zx1≥(m+n)23Nguyễn Văn Quý Geometry Mathley 10.4 2n incenters collinear
Let A1A2A3...An be a bicentric polygon with n sides. Denote by Ii the incenter of triangle Ai−1AiAi+1,Ai(i+1) the intersection of AiAi+2 and Ai−1Ai+1,Ii(i+1) is the incenter of triangle AiAi(i+1)Ai+1 (i=1,n). Prove that there exist 2n points I1,I2,...,In,I12,I23,....,In1 on the same circle.Nguyễn Văn Linh
Geometry Mathley 10.3 2 lines and a circle concurrent
Let ABC be a triangle inscribed in a circle (O). d is the tangent at A of (O),P is an arbitrary point in the plane. D,E,F are the projections of P on BC,CA,AB. Let DE,DF intersect the line d at M,N respectively. The circumcircle of triangle DEF meets CA,AB at K,L distinct from E,F. Prove that KN meets LM at a point on the circumcircle of triangle DEF.Trần Quang Hùng Geometry Mathley 10.2 concurrent, collinear radical centers
Let ABC be an acute triangle, not isoceles triangle and (O),(I) be its circumcircle and incircle respectively. Let A1 be the the intersection of the radical axis of (O),(I) and the line BC. Let A2 be the point of tangency (not on BC) of the tangent from A1 to (I). Points B1,B2,C1,C2 are defined in the same manner. Prove that
(a) the lines AA2,BB2,CC2 are concurrent.
(b) the radical centers circles through triangles BCA2,CAB2 and ABC2 are all on the line OI.Lê Phúc Lữ Geometry Mathley 10.1 right angle wanted
Let ABC be a triangle with two angles B,C not having the same measure, I be its incircle, (O) its circumcircle. Circle (Ob) touches BA,BC and is internally tangent to (O) at B1. Circle (Oc) touches CA,CB and is internally tangent to (O) at C1. Let S be the intersection of BC and B1C1. Prove that ∠AIS=90o.Nguyễn Minh Hà Geometry Mathley 9.3 concyclic, collinear midpoints
Let ABCD be a quadrilateral inscribed in a circle (O). Let (O1),(O2),(O3),(O4) be the circles going through (A,B),(B,C),(C,D),(D,A). Let X,Y,Z,T be the second intersection of the pairs of the circles: (O1) and (O2),(O2) and (O3),(O3) and (O4),(O4) and (O1).
(a) Prove that X,Y,Z,T are on the same circle of radius I.
(b) Prove that the midpoints of the line segments O1O3,O2O4,OI are collinear.Nguyễn Văn Linh Geometry Mathley 9.2 collinear
Let ABDE,BCFZ and CAKL be three arbitrary rectangles constructed outside a triangle ABC. Let EF meet ZK at M, and N be the intersection of the lines through F,Z perpendicular to FL,ZD. Prove that A,M,N are collinear.Kostas Vittas Geometry Mathley 9.1 4 collinear orthocenters
Let ABC be a triangle with (O),(I) being the circumcircle, and incircle respectively. Let (I) touch BC,CA, and AB at A0,B0,C0 let BC,CA, and AB intersect B0C0,C0A0,A0Bv at A1,B1, and C1 respectively. Prove that OI passes through the orthocenter of four triangles A0B0C0,A0B1C1,B0C1A1,C0A1B1.Nguyễn Minh Hà Geometry Mathley 8.4 concurrent circles
Let ABC a triangle inscribed in a circle (O) with orthocenter H. Two lines d1 and d2 are mutually perpendicular at H. Let d1 meet BC,CA,AB at X1,Y1,Z1 respectively. Let A1B1C1 be a triangle formed by the line through X1 perpendicular to BC, the line through Y1 perpendicular to CA, the line through Z1 perpendicular perpendicular to AB. Triangle A2B2C2 is defined in the same manner. Prove that the circumcircles of triangles A1B1C1 and A2B2C2 touch each other at a point on (O).Nguyễn Văn Linh Geometry Mathley 8.3 concyclic
Let ABC be a scalene triangle, (O) and H be the circumcircle and its orthocenter. A line through A is parallel to OH meets (O) at K. A line through K is parallel to AH, intersecting (O) again at L. A line through L parallel to OA meets OH at E. Prove that B,C,O,E are on the same circle.Trần Quang Hùng
Geometry Mathley 8.2 line tangent wanted
Let ABC be a triangle, d a line passing through A and parallel to BC. A point M distinct from A is chosen on d. I is the incenter of triangle ABC,K,L are the the points of symmetry of M about IB,IC. Let BK meet CL at N. Prove that AN is tangent to circumcircle of triangle ABC.Đỗ Thanh Sơn Geometry Mathley 8.1 line bisects segment
Let ABC be a triangle and ABDE,BCFZ,CAKL be three similar rectangles constructed externally of the triangle. Let A′ be the intersection of EF and ZK,B′ the intersection of KZ and DL, and C′ the intersection of DL and EF. Prove that AA′ passes through the midpoint of the line segment B′C′.Kostas Vittas Geometry Mathley 7.3 tangential wanted and given
Let ABCD be a tangential quadrilateral. Let AB meet CD at E,AD intersect BC at F. Two arbitrary lines through E meet AD,BC at M,N,P,Q respectively (M,N∈AD, P,Q∈BC). Another arbitrary pair of lines through F intersect AB,CD at X,Y,Z,T respectively (X,Y∈AB,Z,T∈CD). Suppose that d1,d2 are the second tangents from E to the incircles of triangles FXY,FZT,d3,d4 are the second tangents from F to the incircles of triangles EMN,EPQ. Prove that the four lines d1,d2,d3,d4 meet each other at four points and these intersections make a tangential quadrilateral.Nguyễn Văn Linh Geometry Mathley 7. 2 IK / IO= r / R
A non-equilateral triangle ABC is inscribed in a circle Γ with centre O, radius R and its incircle has centre I and touches BC,CA,AB at D,E,F, respectively. A circle with centre I and radius r intersects the rays [ID),[IE),[IF) at A′,B′,C′. Show that the orthocentre K of △A′B′C′ is on the line OI and that IOIK=RrMichel Bataille
. Geometry Mathley 7.1 EK = EL
Let ABCD be a cyclic quadrilateral. Suppose that E is the intersection of AB and CD,F is the intersection of AD and CB,I is the intersection of AC and BD. The circumcircles (FAB),(FCD) meet FI at K,L. Prove that EK=ELNguyễn Minh Hà Geometry Mathley 6.3 tangent circles
Let AB be an arbitrary chord of the circle (O). Two circles (X) and (Y) are on the same side of the chord AB such that they are both internally tangent to (O) and they are tangent to AB at C,D respectively, C is between A and D. Let H be the intersection of XY and AB,M the midpoint of arc AB not containing X and Y . Let HM meet (O) again at I. Let IX,IY intersect AB again at K,J. Prove that the circumcircle of triangle IKJ is tangent to (O).Nguyễn Văn Linh
Geometry Mathley 6.2 CCC contrsuction, tangent circles
Let ABC be an acute triangle, and its altitudes AX,BY,CZ concurrent at H. Construct circles (Ka),(Kb),(Kc) circumscribing the triangles AYZ,BZX,CXY . Construct a circle (K) that is internally tangent to all the three circles (Ka),(Kb),(Kc). Prove that (K) is tangent to the circumcircle (O) of the triangle ABC.Đỗ Thanh Sơn Geometry Mathley 5.4 5 circles concurrent
Let ABC be a triangle inscribed in a circle (O). Let P be an arbitrary point in the plane of triangle ABC. Points A′,B′,C′ are the reflections of P about the lines BC,CA,AB respectively. X is the intersection, distinct from A, of the circle with diameter AP and the circumcircle of triangle AB′C′. Points Y,Z are defined in the same way. Prove that five circles (O),(AB′C′), (BC′A′),(CA′B′),(XYZ) have a point in common.Nguyễn Văn Linh Geometry Mathley 5.3 3 lines concurrent with Euler line
Let ABC be an acute triangle, not being isoceles. Let ℓa be the line passing through the points of tangency of the escribed circles in the angle A with the lines AB,AC produced. Let da be the line through A parallel to the line that joins the incenter I of the triangle ABC and the midpoint of BC. Lines ℓb,db,ℓc,dc are defined in the same manner. Three lines ℓa,ℓb,ℓc intersect each other and these intersections make a triangle called MNP. Prove that the lines da,db and dc are concurrent and their point of concurrency lies on the Euler line of the triangle MNP.Lê Phúc Lữ Geometry Mathley 5.2 1/ PQ^2 >= 1/ UV^2 - / AC^2
Let ABCD be a rectangle and U,V two points of its circumcircle. Lines AU,CV intersect at P and lines BU,DV intersect at Q, distinct from P. Prove that PQ21≥UV21−AC21
Michel Bataille Geometry Mathley 5.1 fixed line
Let a,b be two lines intersecting each other at O. Point M is not on either a or b. A variable circle (C) passes through O,M intersecting a,b at A,B respectively, distinct from O. Prove that the midpoint of AB is on a fixed line.Hạ Vũ Anh Geometry Mathley 4.4 4 euler lines concurrent
Let ABC be a triangle with E being the centre of its Euler circle. Through E, construct the lines PS,MQ,NR parallel to BC,CA,AB (R,Q are on the line BC,N,P on the line AC,M,S on the line AB). Prove that the four Euler lines of triangles ABC,AMN,BSR,CPQ are concurrent.Nguyễn Văn Linh Geometry Mathley 4.3 concurrent lines, tangent and concurrent circles
Let ABC be a triangle not being isosceles at A. Let (O) and (I) denote the circumcircle and incircle of the triangle. (I) touches AC and AB at E,F respectively. Points M and N are on the circle (I) such that EM∥FN∥BC. Let P,Q be the intersections of BM,CN and (I). Prove that
i) BC,EP,FQ are concurrent, and denote by K the point of concurrency.
ii) the circumcircles of triangle BPK,CQK are all tangent to (I) and all pass through a common point on the circle (O).Nguyễn Minh Hà Geometry Mathley 4.1 P(1, 2)P(3, 4) = P(1, 4)P(2, 3) = P(1, 3)P(2, 4)
Five points Ki,i=1,2,3,4 and P are chosen arbitrarily on the same circle. Denote by P(i,j) the distance from P to the line passing through Ki and Kj . Prove that P(1,2)P(3,4)=P(1,4)P(2,3)=P(1,3)P(2,4)
Bùi Quang Tuấn Geometry Mathley 3.4 tangent circles wanted
A triangle ABC is inscribed in the circle (O,R). A circle (O′,R′) is internally tangent to (O) at I such that R<R′. P is a point on the circle (O). Rays PA,PB,PC meet (O′) at A1,B1,C1. Let A2B2C2 be the triangle formed by the intersections of the line symmetric to B1C1 about BC, the line symmetric to C1A1 about CA and the line symmetric to A1B1 about AB. Prove that the circumcircle of A2B2C2 is tangent to (O).Nguyễn Văn Linh Geometry Mathley 3.3 collinear, 4 lines concurrent
A triangle ABC is inscribed in circle (O). P1,P2 are two points in the plane of the triangle. P1A,P1B,P1C meet (O) again at A1,B1,C1 . P2A,P2B,P2C meet (O) again at A2,B2,C2.
a) A1A2,B1B2,C1C2 intersect BC,CA,AB at A3,B3,C3. Prove that three points A3,B3,C3 are collinear.
b) P is a point on the line P1P2.A1P,B1P,C1P meet (O) again at A4,B4,C4. Prove that three lines A2A4,B2B4,C2C4 are concurrent.Trần Quang Hùng Geometry Mathley 3.2 fixed point
Given a triangle ABC, a line δ and a constant k, distinct from 0 and 1,M a variable point on the line δ. Points E,F are on MB,MC respectively such that MBME=MCMF=k. Points P,Q are on AB,AC such that PE,QF are perpendicular to δ. Prove that the line through M perpendicular to PQ has a fixed point.Nguyễn Minh Hà
Geometry Mathley 3.1 OI = OJ
AB,AC are tangent to a circle (O), B,C are the points of tangency. Q is a point iside the angle BAC, on the ray AQ, take a point P suc that OP is perpendicular to AQ. The line OP meets the circumcircles triangles BPQ and CPQ at I,J. Prove that OI=OJ.Hồ Quang Vinh Geometry Mathley 2.4 concurrency
Let ABC be a triangle inscribed in a circle of radius O. The angle bisectors AD,BE,CF are concurrent at I. The points M,N,P are respectively on EF,FD, and DE such that IM,IN,IP are perpendicular to BC,CA,AB respectively. Prove that the three lines AM,BN,CP are concurrent at a point on OI.Nguyễn Minh Hà Geometry Mathley 2.3 two lines and a circle concurrent
Let ABC be a triagle inscribed in a circle (O). A variable line through the orthocenter H of the triangle meets the circle (O) at two points P,Q. Two lines through P,Q that are perpendicular to AP,AQ respectively meet BC at M,N respectively. Prove that the line through P perpendicular to OM and the line through Q perpendicular to ON meet each other at a point on the circle (O).Nguyễn Văn Linh Geometry Mathley 2.2 concurrency
Let ABC be a scalene triangle. A circle (O) passes through B,C, intersecting the line segments BA,CA at F,E respectively. The circumcircle of triangle ABE meets the line CF at two points M,N such that M is between C and F. The circumcircle of triangle ACF meets the line BE at two points P,Q such that P is betweeen B and E. The line through N perpendicular to AN meets BE at R, the line through Q perpendicular to AQ meets CF at S. Let U be the intersection of SP and NR,V be the intersection of RM and QS. Prove that three lines NQ,UV and RS are concurrent.Trần Quang Hùng Geometry Mathley 2.1 OM >= R , S_bS_c = S_aS_b+S_aS_c
Let ABC be an equilateral triangle with circumcircle of center O and radius R. Point M is exterior to the triangle such that SbSc=SaSb+SaSc, where Sa,Sb,Sc are the areas of triangles MBC,MCA,MAB respectively. Prove that OM≥R.Nguyễn Tiến Lâm Geometry Mathley 1.4 concurrency of 4 lines
Given are three circles (O1),(O2),(O3), pairwise intersecting each other, that is, every single circle meets the other two circles at two distinct points. Let (X1) be the circle externally tangent to (O1) and internally tangent to the circles (O2),(O3), circles (X2),(X3) are defined in the same manner. Let (Y1) be the circle internally tangent to (O1) and externally tangent to the circles (O2),(O3), the circles (Y2),(Y3) are defined in the same way. Let (Z1),(Z2) be two circles internally tangent to all three circles (O1),(O2),(O3). Prove that the four lines X1Y1,X2Y2,X3Y3,Z1Z2 are concurrent.Nguyễn Văn Linh Geometry Mathley 1.3 collinear
Let ABC be an acute triangle with incenter O, orthocenter H, altitude AD.AO meets BC at E. Line through D parallel to OH meet AB,AC at M,N, respectively. Let I be the midpoint of AE, and DI intersect AB,AC at P,Q respectively. MQ meets NP at T. Prove that D,O,T are collinear.Trần Quang Hùng Geometry Mathley 1.1 hexagon perimeter inequality
Let ABCDEF be a hexagon having all interior angles equal to 120o each. Let P,Q,R,S,T,V be the midpoints of the sides of the hexagon ABCDEF. Prove the inequality p(PQRSTV)≥23p(ABCDEF), where p(.) denotes the perimeter of the polygon.Nguyễn Tiến Lâm