geometry problems from Argentinian Training Lists for Cono Sur, Rioplatense, IberoAmerican with aops links in the names
collected inside aops: here
Cono Sur: 1993, 2013-14, 2018
IberoAmerican: 1994-99, 2019
Rioplatense: 2019
Cono Sur 2013 , 2014 , 2018
In triangle ABC, let D, M be interior points of sides BC and AD , respectively. Lines BM and AC intersect at E , lines CM and AB intersect at F , and lines EF and AD intersect at N . Prove that2\frac{AN}{DN}=\frac{AM}{DM}.
Let I be the incenter of triangle ABC . The inscribed circle is tangent to sides BC , CA and AB and points A', B' and C' respectively. Let M be the midpoint of the segment A'B'. Let P be the projection of I on AA'. The straight lines MP and AC meet at N . Prove that the lines A'N and B'C' are parallel.
Let ABC be an acute triangle. We consider the points M , N in the interior of the side BC , Q in the interior of the side AB and P in the interior of the side AC, such that MNPQ is a rectangle. Prove that if the center of the rectangle MNPQ is also the center of the triangle ABC , then AB=AC=3AP.
Let ABCD be a square and E be a point on the side AB . Lines DE , BC meet at F, and lines CE , AF meet at G. Prove that the lines BG and DF are perpendicular.
Let ABCD be a square and P be an interior point such that PA=1 , PB=\sqrt2 and PC=\sqrt3.
a) Find the length of PD
b) Find the measure of the angle \angle APB
Let ABC be a triangle with right angle A . We consider the points D in the interior of the side AC and E in the interior of the side BD , so that \angle ABC=\angle ECD=\angle CED . Provethat BE=2AD.
Let ABCD be a quadrilateral inscribed in a circle with center O . Lines AC , BD meet at P and lines AB , CD meet at Q . Let R be the second intersection point of the circumscribed circles of the triangles ABP and CDP.
a) Prove that P , Q and R are collinear.
b) Let U and V be the circumcenters of the triangles ABP and CDP , respectively. Prove that the points U , R , O , V are concyclic.
In the triangle with acute angle ABC, with AB\ne AC. Let D be the foot of the bisector of the angle \angle A. Let E,F be the feet of the altitudes from B and C, respectively. The circles circumscribed by the triangles DBF and DCE meet for second time at M. Demonstrate that ME=MF.
Let H be the orthocenter of the acute triangle ABC and P a point on the circumscribed circle of triangle ABC. Show that the Simson Line of P divides the segment PH in half.
The incircle of triangle ABC is tangent to AB and AC at D and E respectively. Let X be the point on the line DE such that DE is tangent to the circle that passes through B, C and X. Prove that the angle \angle BXC is obtuse.
Let ABCD be a cyclic quadrilateral and \omega_1, \omega_2 the circles inscribed in the triangles ABC and BCD. Show that the common exterior tangent to \omega_1 and \omega_2, other than BC, is parallel to AD.
We consider a circle with center O and radius r and a line \ell that does not pass through O. A cricket jumps between the points of the line and those of the circle, with jumps of length r. Show that there are at most 8 points that the cricket can reach.
Let A be a point on the semicircle of diameter BC, and X an arbitrary point inside the triangle ABC. Line BX intersects semicircle a second time at K, and intersects segment AC at F. Line CX intersects semicircle for second time at L, and intersects segment AB at E. Prove that the circumscribed circles of the triangles AKF and AEL are tangent.
Let D be the midpoint of side BC of triangle ABC with AB \ne AC and E the foot of the altitude BC. If P is the intersection point of the perpendicular bisector of segment DE with the perpendicular from D on the bisector of \angle BAC, prove that P lies on the Euler circle of triangle ABC.
Let ABC be a triangle and let Z be a point inside it that does not belong to the perimeter of ABC. Let D be the intersection of the line AZ with BC. Let X be a point in AB such that XD bisects segment BZ. Similarly let Y be a point in AC such that YD bisects segment CZ. Show that Z lies in the interior of triangle AXY.
Let ABC be a triangle with right angle at A. The altitude from A intersects BC at H and M is the midpoint of hypotenuse BC. On the legs and outside the triangle, the equilateral triangles BAP and ACQ are constructed. If N is the intersection point of the lines AM and PQ, show that the angles \angle NHP and \angle AHQ are equal.
We have the right triangle ABC with \angle ABC = 90^o and we choose points D and E in the segments AB and BC respectively such that BD = BE. Let G be the point such that DG\perp AB and EG\perp BC and let F be the intersection of lines AE and CD. If M is the midpoint of AC, show that FG and BM intersect at a point on the circumscribed circle of ABC.
Danielle practices billiards at a circular table. He chooses two points P_0 and P_1 on the perimeter of the table, places the ball at point P_0 and hits it towards point P_1. By doing this, he realizes that the ball bounces n times at different points P_1, P_2,..., P_n where P_n = P_0. Show that the points P_0, P_1,...,P_{n - 1} are the vertices of a regular polygon.
Let ABCD be a convex quadrilateral with AB\ge CD. On the segment AB, points E and F are chosen and on the segment CD points G and H are chosen such that AE = BF = CG = DH <\frac{AB}{2}. Let P, Q and R be the midpoints of EG, FH and CD respectively. It is known that PR is parallel to AD and QR is parallel to BC.
a) Show that ABCD is a trapezoid.
b) Let d be the difference of the lengths of the parallel sides. Show that 2 PQ \le d.
Let P be an interior point of the acute triangle ABC. Show that if the symmetrics of P with respect to the sides of the triangle belong to the circumcircle of the triangle, then P is the orthocenter of ABC.
Let ABC be a nondegenerate triangle and I the point of intersection of its angle bisectors. D is the midpoint of the segment CI . If it is known that \frac{AD}{DB} =\frac{BC}{AC} , show that AC = BC.
Two different circles \omega_1 and \omega_2 such that neither contains the other intersect at A and B. Let C be a point at \omega_2 (C \ne A and C \ne B). Lines CA and CB intersect \omega_1 again at F and G respectively. The tangents to \omega_1 through F and G intersect at D. The lines AG and BF intersect at E. Prove that C, D and E are collinear.
Let ABC be an acute triangle, \Gamma the circle circumscribed to triangle ABC and D a point on segment BC. Let M be the midpoint of AD, E the foot of the perpendicular from D on AB and F the intersection of the line DE with the small arc BC of \Gamma. If it is known that \angle DAE = \angle AFE, show that the lines EM and CF and the tangent line through A to \Gamma concur.
Let ABCDEF be a convex hexagon with non-parallel sides and tangent to a circle \Gamma at the midpoints P, Q, R of the sides AB, CD, EF respectively. \Gamma is tangent to BC, DE and FA at points X, Y, Z respectively. Line AB intersects lines EF and CD at points M and N respectively. Lines MZ and NX intersect at point K. Let r be the line joining the center of \Gamma and K. Show that the seven lines AD, PY, BE, RX, r, CF and QZ have a point in common.
Given an acute triangle ABC we construct exterior triangles ABD and ACE such that \angle ADB =\angle AEC = 90^o and \angle BAD = \angle CAE. Let A_1\in BC, B_1\in AC, C_1\in AB be the feet of the heights of triangle ABC, and let K and L be the midpoints of BC_1 and CB_1 respectively. Show that the circumcenters of the triangles AKL, A_1B_1C_1 and DEA_1 are collinear.
The inscribed circumference of triangle ABC touches sides BC, CA, AB at points D, E, F respectively. In the segments EF, FD, DE we consider the points M, N, P respectively such that BM + MC, CN + NA, AP + PB are minimal.
\bullet Prove that the lines AM, BN, CP are concurrent.
\bullet Prove that DM, EN, FP are altitudes of the triangle DEF.
Let I be the incenter of the scalene triangle ABC, with AB <AC, and let I' be the symmetric of I wrt BC. The bisector AI intersects the side BC at D and the circumscribed circle of ABC at E. The line EI' intersects the circumscribed circle of ABC at F. Show that \frac{AI}{IE} = \frac{ID}{DE} and IA = IF.
Rioplatense L2,L3 2019
Let ABCD be a cyclic quadrilateral with AB = AD. Two points M and N interior of the segments CD and BC respectively are such that DM + BN = MN. Show that the circumcenter of triangle AMN lies on segment AC.
Let C_1 and B_1 be points on the sides AB and AC of triangle ABC respectively. Segments BB_1 and CC_1 intersect at X and segments B_1C_1 and AX intersect at A_1. The circumscribed circles of triangles BXC_1 and CXB_1 intersect side BC at D and F respectively. Lines B_1D and C_1E intersect at F. Show that the lines A_1F, B_1E, C_1D are parallel or concurrent.
Let ABCD be a convex quadrilateral. Point A_1 belongs to the edge of ABCD and is such that segment AA_1 divides quadrilateral ABCD into two parts of equal area. In the same way, points B_1, C_1 and D_1 are defined. The lengths of segments AA_1, BB_1, CC_1, and DD_1 are known to be less than or equal to 1. Show that area \, (ABCD) <\frac23.
A point X is located in a right isosceles triangle ABC with \angle B = 90^o. Prove the inequality AX + BX+\sqrt2 CX \ge \sqrt5 ABand find all points X for which equality applies.
IberoAmerican 1994-1999, 2019
Let ABC be a triangle such that \angle A <\angle A <90^o <\angle B. Let P be the intersection of the external bisector of \angle A with the line BC and Q the intersection of the external bisector of \angle B with the line AC. Knowing that AP = BQ = AB, calculate \angle A.
Let a, b, and c be the lengths of the sides of a triangle and let p the semiperimeter and r the inradius of the triangle. Prove that
\frac{1}{(p-a)^2}+\frac{1}{(p - b)^2}+\frac{1}{(p-c)^2}\ge \frac{1}{r^2}
If A, B, C and D are four different points such that every circle passing through A and B has nonempty intersection with every circle passing through C and D, show that the four points are collinear or concyclic.
Let ABCDEF be a hexagon inscribed in a circle of radius r. Prove that if AB = CD = EF = r, then the midpoints of BC, DE and FA are vertices of an equilateral triangle.
Let P be a convex polygon contained in a square of side 1. Prove that the sum of the squares of the sides of P is less than or equal to 4.
Construct a triangle ABC if the points M, N and P in AB, AC and BC respectively , are known and verify the property\frac{MA}{MB} = \frac{PB}{PC} = \frac{NC}{CA} = kwhere k is a fixed number between 0 and 1.
Find all right triangles with integer sides such that their area is twice their perimeter.
Let A, B, C be three points of the plane with the two integer coordinates and let R be the radius of the circle that contains A, B and C. If a, b and c are the lengths of the sides of triangle ABC, prove that abc \ge 2R.
Let ABCD be an cyclic quadrilateral, M the orthocenter of triangle ABC, and N the orthocenter of triangle ABD. Show that MNDC is a parallelogram.
Let P_1P_2P_3P_4P_5 be a non-intersecting flat pentagon, totally contained between the line r that passes through P_1 and P_5 and the line s, parallel to r passing through P_3. Let a> 0. Prove that it is possible to choose points P_6 and P_7 such that P_6P_2 = a, so that it is possible to pave the plane with mosaics congruent to the heptagon P_1P_2P_3P_4P_5P_6P_7.
Consider the triangle ABC with AB <AC.
a) Show that there are points M, N, such that B \in MC, C \in NB and the inradii of the triangles AMB, ABC, and ACN are equal.
b) If M, N, are as in (a), show that MB <NC and \angle MAB> \angle NAC.
Show that if an octahedron has all its faces triangular and these faces are parallel in pairs, then the faces are equal in pairs.
Let ABC be a triangle and A', B', C' points on the sides BC, CA and AB respectively. Prove that if the lines AA', BB' and CC' have a point in common P, with P \ne A and P \ne A ', then
\frac{B'C}{B'A} + \frac{C'B }{C'A} \ge 4 \frac{PA'}{PA}.Find the locus of the points P for which the equality holds.
Prove that a triangle with prime sidelengths and integer area doesn't exist.
In right triangle ABC, with \angle C = 90^o, let D be a point on side AB and M be the midpoint of CD. If \angle AMD = \angle BMD, show that\frac{\angle ACD}{\angle BCD}=\frac{\angle CDA}{\angle CDB}.
Let O be the point of intersection of diagonals AC and BD of quadrilateral ABCD. Prove that the orthocenters of the four triangles OAB, OBC, OCD, ODA are vertices of a parallelogram that is similar to the one determined by the centroids of the same triangles.
Let AB and AC be tangent to the circle \Gamma at B and C respectively. Let D be a point on the extension of AB, beyond B, and let P be the second intersection point of \Gamma with the circle circumscribed to the triangle ACD. Let Q be the foot of the perpendicular from B to CD. Prove that \angle DPQ =2 \angle ADC.
Let ABC be a triangle and M be a point. We consider the circles of diameters AM, BM, CM and the circle that contains these three and is tangent to all three. Show that the radius P of this large circle satisfies P\ge 2r, where r is the inradius of ABC.
You want to decorate a rectangular box with a ribbon that passes through its faces and forms various angles with the edges. If possible, the points where the tape crosses the edges are chosen so that the length of the closed path is minimal (local). This ensures that the ribbon can be adjusted and tied without slipping. Decide if there is a minimum path that crosses all 6 faces exactly once.
Quadrilateral ABCD is inscribed in a circle with center O. AD and BC intersect at P. Let L and M be the midpoints of AD and BC respectively, and let Q and R be the feet of the perpendiculars from O and P on LM. Prove that LQ = RM.
Let a, b, c be the sides and \alpha, \beta,\gamma and the angles (in radians) of a triangle of semiperimeter s. Prove that
\frac{bc}{\alpha(s-a)}+ \frac{ca}{\beta(s-b)}+\frac{ab}{\gamma(s-c)} \ge \frac{12s}{\pi}
Let E and F be points on sides BC and AD, respectively, of quadrilateral ABCD. Let P be the intersection of AE with BF and Q the intersection of CF with DE. Show that E and F divide BC and AD (or BC and DA) in the same ratio if and only if\frac{area\,\, (FPQ)}{area \,\,(EDA)}= \frac{area \,\, (EQP) }{area\,\, (FBC)}
In a triangle ABC, let S be the midpoint of the median corresponding to vertex A and Q the intersection point of BS with the side AC . Show that BS = 3QS.
Let A_1A_2... A_n be an n-sided polygon with centroid G, inscribed in a circle. The lines A_1G, A_2G, ..., A_nG intersect again the circle at B_1, B_2, ..., B_n, respectively. Prove that
\frac{A_1G}{GB_1}+\frac{A_2G}{GB_2}+...+\frac{A_nG}{GB_n}=n
Let ABC be a right triangle with the right angle at C and with legs BC = a and CA = b. Let D and E be points on sides AC and AB respectively, so that BD is the bisector of angle B and DE is parallel to BC. Prove that the area of the triangle ADE, (ADE), satisfies the inequality(ADE)< \frac{a^4+b^4}{2(a+b)^2}
Let ABC be a triangle, G be the centroid, and P be a variable point in the interior of ABC. Let D, E, F be points on the sides BC, CA, AB respectively, such that PD\parallel AG, PE\parallel BG, PF\parallel CG. Prove that the sum of areas (PAF) + (PBD) + (PCE) is constant.
In a plane consider the equilateral triangles ABC and A'BC. Let D be a variable point on side AC. Line A'D intersects line AB at E. Line BD intersects line CE at P. Show that P lies on the circle circumscribed to triangle ABC.
Triangle ABC is right at C.
a) Show that the three ellipses that have foci at two vertices of the triangle and pass through the third have a point in common.
b) Show that the principal vertices of the ellipses in part (a), that is, the points at which each ellipse intersects the line joining its foci, form two collinear point themes.
Given a regular tetrahedron of height 1, find the volume that will remain after removing, in a first step, the inscribed sphere; then, in a second step, the following spheres relative to each of the 4 vertices, externally tangent to the inscribed sphere; and in the subsequent steps, following the process without interruption, the following 4 spheres relative to the vertices, each tangent externally to the sphere, relative to the same vertex, created in the previous step.
Clarification. Given a sphere tangent to the internal faces adjacent to a vertex of a polyhedron we will call the following sphere relative to the vertex to the sphere tangent externally to the first sphere and tangent internally to the same faces of the polyhedron.
The triangle ABC is inscribed in the circle \Gamma. Let AA_1, BB_1, CC_1 be the bisectors of the angles \angle A, \angle B, \angle C with A_1, B_1, C_1 in \Gamma. Prove that the perimeter of the triangle is equal to
AA_1 \cos \left(\frac{\angle A}{2}\right)+BB_1 \cos \left(\frac{\angle B}{2}\right)+CC_1 \cos \left(\frac{\angle C}{2}\right)
Let M be a variable point on side BC of triangle ABC. A line through M intersects lines AB and AC at K and L respectively, such that M is the midpoint of KL. Point K' is such that ALKK' is a parallelogram. Determine the locus of K' as M moves in segment BC.
Let ABCD be a square and E, F be points on the sides BC, CD, respectively, such that CE\ne CF. Lines AE and AF intersect BD at P and Q, respectively. It is assumed that BP \cdot CE = DQ\cdot CF. Prove that the five points C, E, F, P and Q lie on a circle.
Let P be an interior point of the equilateral triangle ABC such that PB\ne PC. Let D be the intersection point of BP with AC and E be the intersection point of PB with AD and E be the intersection point of CP with AB. Assuming that \frac{PB}{PC}= \frac{AD}{AE}, find measure of the angle \angle BPC .
Let ABC be an acute triangle of orthocenter H and circumcenter O. The perpendicular bisector of segment AH intersects AB at P and AC at Q. Prove that \angle AOP = \angle AOQ.
Let ABC be a triangle with \angle A> 90^o and AD, BE, CF its altitudes . Let E', F' be the feet of the perpendiculars from E and F on BC. Assuming 2E'F'= 2AD + BC, find \angle A.
An isosceles triangle has integer sides; furthermore, the length of the base and the length of the other two sides are coprime. If the angle bisectors have rational lengths, show that the length of the equal sides is an odd perfect square.
Let ABC be a non-equilateral triangle of circumcenter O and incenter I. Let D be the foot of the altitude drawn from A. If the circumradius R is equal to the radius r_a of the exscribed circle corresponding to BC, show that O, I, and D are collinear.
Let a, b and c be the lengths of the sides of a scalene triangle \Delta and let l =\frac{b + c}{2}, m = \frac{c + a}{2} and n =\frac{a + b}{2}. Prove that l, m and n are lengths of the sides of a scalene triangle whose area is greater than that of triangle \Delta.
Let ABCD be a trapezoid with the property that the triangles ABC, ACD, ABD and BCD have the same inradius. Prove that A, B, C and D are the vertices of a rectangle.
A parallelogram is inscribed in a regular hexagon in such a way that the centers of symmetry of both figures coincide. Prove that the area of the parallelogram is less than or equal to 2/3 of the area of the hexagon.
Let ABC be a non-obtuse triangle. For each point P in segment BC let Q in segment AC and R in segment AB, such that triangle PQR has minimum perimeter. Prove that if all the lines QR are concurrent (when P moves along \overline{BC}) then the angle \angle BAC is right.
Let P be a point in the interior of a circle, other than the center. Three lines are drawn through P at angles of 60^o, so that none of them is diameter; these divide the circle into 6 regions. If we shade 3 of the 6 sectors thus determined alternately around P, we will have two regions within the circle: one shaded and the other unshaded. Prove that of these two regions, the one with the greatest area is the one that contains the center of the circle.
A circle \Gamma of radius R is tangent to a line t at a point C. Another circle \omega of radius r <R is tangent to t at a point B and intersects \Gamma at points A and A', A being closest to t. What is the locus of the centers of the circles circumscribed to the triangles ABC when the circle \omega rolls on the line t?
In a convex polygon P, some diagonals have been drawn that do not intersect inside P. Show that there are at least two vertices of P$$ such that neither of them is an endpoint of any of the diagonals drawn.
On the sides AB and BC of the square ABCD, the points E and F were chosen, respectively, such that BE = BF. Let BN be the altitude of triangle BCE. Prove that DNF is a right angle.
The centers of the spheres circumscribed and inscribed to a given tetrahedron coincide. Prove that the faces are congruent triangles.
Let P be a point in the interior of a tetrahedron ABCD. Show that the sum of the angles APB, APC, APD, BPC, BPD, CPD is greater than 540^o
Let A_1A_2A_3 be a triangle ,points B_1, B_2, B_3 be on the sides A_2A_3, A_3A_1, A_1A_2, respectively (which do not coincide with any vertex). Show that the perpendicular bisectors of the three segments A_1B_1 are never concurrent
Given four points in space A, B, C, D. Let M and N be the midpoints of AC and BD respectively. Prove that
AB^2 + BC^2 + CD^2 + DA^2 = AC^2 + BD^2 + 4MN^2.
Through the ends A and B of a diameter AB of a given circle, the tangents \ell and m are drawn, parallel to each other. Let C be a point on \ell other than A and let q_1, q2 be two rays with origin at C. Suppose that q_i (i = 1, 2) intersects the circle at points D_i, E_i (with D_1 between C and E_i). Rays AD_1, AD_2, AE_1, AE_2 intersect m at points M_1, M_2, N_1, N_2, respectively. Prove that M_1M_2 = N_1N_2.
In triangle ABC, let r be the inradius and let r_A be the radius of the circle tangent to AB, to AC and externally tangent to the circle inscribed at ABC, r_B and r_C are defined similarly. Prove that
r_A + r_B + r_C\ge rand the equality holds if and only if the triangle is equilateral.
In a tetrahedron ABCD the foot of the height drawn from each vertex coincides with the incenter of the opposite face. Prove that the tetrahedron is regular.
In a sphere of radius 1, consider a maximum circle H. On one of the hemispheres determined by H consider three circles of equal radii, tangent to each other, in pairs, and tangent to H. Determine the radius of another circle, located on the same hemisphere and that is tangent to the three circles.
A lattice triangle has the following property: "the product of the lengths of two of its sides is a prime number and its area is a prime number". Find the area and determine all lattice triangles that have the above property.
Consider a square ABCD and the arbitrary points P and Q on the sides AB and BC, respectively. Lines AQ and PC intersect PD and QD at points X and Y. Show that lines AY, CX, and BD are concurrent.
Let A_1A_2A_3 be a non-right triangle. The points O_1, O_2, O_3 are the centers of the circles \Gamma_1, \Gamma_2, \Gamma_3 tangent in pairs (internally or externally); the circle \Gamma_1 passes through A_2 and A_3, the circle \Gamma_2 passes through A_3 and A_1, the circle \Gamma_3 passes through A_1 and A_2. Knowing that the triangles A_1A_2A_3 and O_1O_2O_3 are similar, determine their angles.
Consider a semicircle of diameter AB = 2R and a chord CD of fixed length c. Let E be the intersection of AC with BD and F the intersection of AD with BC. Show that the segment EF has constant length and is perpendicular to AB. Express EF in terms of R and c.
A regular heptagon A_1A_2... A_7 is inscribed in circle C. Point P is taken on the shorter arc A_7A_1.
Prove that PA_1+PA_3+PA_5+PA_7 = PA_2+PA_4+PA_6.
We consider a circle \Gamma, an exterior point A and a line tangent to \Gamma through A, where T is the point of tangency. A variable line r is drawn through A that cuts the circle \Gamma at P and B in such a way that P belongs to segment AB. For each r we take, in the same half plane as T, the point C on the perpendicular bisector of AB such that the distance from C to r is equal to AB.
a) Determine the locus of C as line r varies.
b) Prove that the area of the triangle APC is constant as line r varies.
Prove that in a convex quadrilateral of area 1 the sum of all the sides and all the diagonals is greater than or equal to 4 +\sqrt8.
Given a convex polygon, prove that we can find a point Q inside the polygon and three vertices A_1, A_2, A_3 (not necessarily consecutive) such that each ray A_iQ (i = 1, 2,3) forms an acute angle with each one of the sides that have a vertex at A_i.
Let P be an interior point of the tetrahedron ABCD and S_A,S_B,S_C,S_D the center of gravity of the tetrahedra PBCD, PCDA, PDAB, PABC. Prove that the volume of the tetrahedron S_AS_BS_CS_D is equal to \frac{1}{64} of the volume of tetrahedron ABCD.
Let M be the set of all tetrahedra whose inscribed and circumscribed spheres are concentric. We denote with r and R the radii of said spheres. Determine the range of values \frac{R}{r} for tetrahedra of M.
Through a point P inside a sphere, draw three chords perpendicular in pairs. Prove that the sum of the squares of their lengths does not depend on their directions.
Let ABCD be a parallelogram. Circle S_1 passes through vertex C and is tangent to sides BA and AD at points P_1 and Q_1 respectively. Circle S_2 passes through vertex B and is tangent to sides DC and AD at points P_2 and Q_2 respectively. Let d_1 and d_2 be the distances from C and B to the lines P_1Q_1 and P_2Q_2 respectively. Find all possible values of the ratio d_1: d_2.
Let ABC be a triangle, I its incenter, \Gamma its circumscribed circle, and M the midpoint of side BC. Suppose that there are two different points P, Q on the line IM such that PB = PI and QC = QI. Let G be the point of intersection of the lines PB and QC. Prove that the line IG and the perpendicular bisector of segment BC intersect at a point on \Gamma.
ABC is a triangle with incenter I. We construct points P and Q such that AB is the bisector of \angle IAP, AC is the bisector of \angle QAI and \angle PBI + \angle IAB = \angle QCI + \angle IAC = 90^o. The lines IP, IQ intersect AB, AC at K, L respectively. The circumscribed circles of the triangles APK and AQL intersect at A and R. Show that R lies on the line joining the midpoints of BC and KL.
Let A_1H_1, A_2H_2 and A_3H_3 be the altitudes and A_1L_1, A_2L_2 and A_3L_3 the angle bisectors of an acute triangle A_1A_2A_3. Prove that area \, (L_1L_2L_3) \ge area \, (H_1H_2H_3).
Let ABC be an acute triangle with AB \ne AC and H its orthocenter. Let D and E be the intersections of BH and CH with AC and AB respectively, and P the foot of the perpendicular from A on DE. The circumcircle of BPC intersects DE at a point Q \ne P. Show that lines AP and QH intersect at a point on the circumcircle of ABC.
In triangle ABC, the inscribed circle \omega is tangent to sides BC, CA , AB at points D, E, F respectively. Let M and N be the midpoints of DE and DF respectively. Points D', E', F' are considered on the line MN such that D'E = D'F, BE' || DF and CF' || DE. Show that the lines DD', EE' and FF' are concurrent.
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