geometry problems from Slovakia Team Selection Tests (TST) with aops links in the names
(only those not in IMO Shortlist)
2000 - 2017
under construction
2000 Slovakia TST p3
Let ABCDEF be a convex hexagon such that | AB | = | BC |, | CD | = | DE |, | EF | = | F A |.
Prove that the (extended) heights of the triangles BCD, DEF, FAB successively from the vertices C, E, A intersect at one point.
2000 Slovakia TST p5
Let P, Q, R be successively the centers of the circular arcs BC, CA, AB of the circle described
given triangle ABC. Let K, L, M further denote successively the centers of its sides BC, CA,
AB and I are the center of the circle inscribed in this triangle. Prove that is true
| AI | · | BI | · | CI | = 8 · | KP | · | LQ | · | MR |.
2000 Slovakia TST p8
Let ABCD be a quadrilateral inscribed in a circle with center O. Let P be an intersection
diagonals AC and BD. The centers of the circles described are the triangles ABP, BCP, CDP and DAP
are O1, O2, O3 and O4, respectively. Prove that the lines OP, O1O3 and O2O4 intersect in one
bode.
2000 Slovakia TST p11
The vertices A, B, C of the acute triangle ABC lie successively on the sides B1C1, C1A1,
A1B1 of triangle A1B1C1 such that | <ABC | = | <A1B1C1 |, | <BCA | = | <B1C1A1 |
a | <CAB | = | <C1A1B1 |. Prove that the orthocenters (intersections of the heights) of the triangles ABC and A1B1C1 are equidistant from the center of the circle of the described triangle ABC
2000 Slovakia TST p14.
All three vertices of a triangle have both integer coordinates, the length of one of the sides is
\ sqrt n, where n is not divisible by any square of the prime number. Prove the radius ratio
the inscribed circle described by this triangle is an irrational number
2000 Slovakia TST p17
Let ABCD be a string quadrilateral and a positive real number k. Let E and F be points respectively
on pages AB and CD also that | AE | : | EB | = | CF | : | F D | = k. Let P be such a point on
the line segment EF that | P E | : | P F | = | AB | : | CD |. Prove that the ratio of the contents of the triangles AP D and BP C does not depend on the number k.
2001 Slovakia TST p1
There are two circles k1 and k2, which intersect at two different points A and B. Line p
passing through point A intersects the circles k1 and k2 respectively at points C and D. Let P and Q be
projections of point B constructed successively on the tangent to the circle k1 at point C and on the tangent to the circle k2 at point D. Prove that the line P Q is a tangent to the circle k3 constructed above
diameter AB.
2001 Slovakia TST p8
Prove that on the surface of the nineteen wall described by a sphere with radius 10, there are two
points whose distance is greater than 21
2001 Slovakia TST p9
Three circles of the same radius equal to t pass through one point T, all inside
triangle ABC and each of them touches two sides of it. Let r be the radius of the inscription and R
radius of the described circle of triangle ABC. Prove that:
(i) t = rR / (R + r);
(ii) T lies on a line passing through the center of the inscribed and the center of the described circle of the triangle ABC
2001 Slovakia TST p12
Prove that for every natural number n >= 6 there exists a set M containing n points in a plane such that for each point P from the set M there exist at least 3 points in M at a distance of 1 from P!
2001 Slovakia TST p16.
Prove that if in the convex pentagon ABCDE holds |< ABC | = |<ADE | a |<AEC | =
= |<ADB |, so |<BAC | = |< DAE |.
2002 Slovakia TST p5
In a given triangle ABC, |<BAC | > |< BCA |. Inside the triangle ABC is given
point P so that |< P AC | = |< BCA |. Outside the triangle ABC lies the point Q so that P Q k AB
and BQ to AC. Let R be a point on the side BC (R is separated from the point Q by the line AP) such that |< P RQ | = |< BCA |. Prove that the circles described by the triangle ABC and P QR are relative to each other they touch.
2002 Slovakia TST p8
in a circle k1 and k2 with the centers in series at points P and Q intersect at points S and T.
The common (outer) tangent of these circles, which is closer to the point S, is denoted by p and its
touch points with circles k1, k2 in series M and L.
(a) The second intersection of the tangent to the circle k2 at the point S with the circle k1 is denoted by K. the lines SL and MK are denoted by U. Prove that the lines MU and MS are tangents to a circle
described triangle ST U.
(b) Let R be the intersection of the axes of the MS and SL sides. The point W is the second intersection of the circle described triangle P QR with line RS. Prove that S is the center of the line segment RW
2002 Slovakia TST p10
The diagonals AC and BD of the convex quadrilateral ABCD intersect at point E. Prove
that inequalities apply to the contents of the departments
\ sqrt SABE + \ sqrt SCDE <= \ sqrt SABCD.
When does equality occur?
2002 Slovakia TST p12
Points A1, A2, A3, A4 are given on the sphere described by a regular tetrahedron with an edge of length 1 also that | AiAj | <1 for each i 6 = j. Prove that these four points lie on one hemisphere.
2002 Slovakia TST p14
There is a triangle ABC in which β <45◦ . On the side BC lies the point D such that the center of the circle of the entered triangle ABD is identical with the center O of the circle of the described triangle ABC. Let 1 be the circle described by triangle AOC. Let P be the intersection of the tangents to the circle l at points A and C, further denote by Q the intersection of the lines AD and CO and finally X let be the intersection the line P Q and the tangent to l at point O. Prove that | XO | = | XD |.
2003 Slovakia TST p3
Find the largest p ∈ R such that for each triangle with sides a, b, c and the content S is
inequality met a ^ 2 + b ^ 2> = pc ^ 2 + 3S.
2003 Slovakia TST p4
The two circles k1 and k2 have an external contact at point K. Both have an internal contact with the circle k respectively at points A1 and A2. Let P be one of the intersections of the circle k with the common by the tangent of the circles k1 and k2 led by the point K. For i = 1, 2, let Bi denote the intersection of the line P Ai with a circle ki (different from Ai). Prove that the line B1B2 is a common tangent of the circles k1 and k2
2003 Slovakia TST p5
In an acute triangle ABC, denote by H the heel of the height drawn from the vertex A. Let P be
any interior point of the triangle ABC and let D, E and Q be the heel of the heights of the heights
led by AB, AC and AH. Prove that is true
| AB | · | AD | - | AC | · | AE | = | BC | · | P Q |.
2003 Slovakia TST p11
A right triangle ABC with right angle at vertex C is given, where | BC | <| AC |.
Let I denote the center of the inscribed circle, O the center of the side AB. On the AC, AB sides, choose one after the other points Q, P so that | CQ | = | CB | = | BP |. Intersection of the angle axis ACB and the line QP denote S. Perpendicular to QI passing through point S intersects BQ at point R. Prove that the points I, R, O lie on one line.
2003 Slovakia TST p13
A sphere is inscribed in the quadrilateral, which touches one wall at the intersection of the heights, the other touches the center of gravity and the third touches the center of the inscribed circle. Prove that then there is a quadrilateral regular
2003 Slovakia TST p19
The two circles k1 and k2 intersect at points P and Q. Let's choose points A1, B1
on circle k1. Lines A1P and B1P intersect the circle k2 except for point P at points A2, B2
and lines A1B1, A2B2 intersect at point C. Prove that when we change the position of points A1, B1,
thus, the centers of the circles described by triangle A1A2C lie on the circle
2004 Slovakia TST p1
There is a triangle ABC and on the side BC a point D such that | AD | > | BC |. Point E on the AC side
is determined by the ratio | AE | / | EC | = | BD | / (| AD | - | BC |).
Prove that | AD | holds > | BE |.
2004 Slovakia TST p5
On the arc BC of the circle of the described triangle ABC, which does not contain the point A, we choose the point P. On the half-lines AP, BP we choose the points X, Y successively so that | AC | = | AX |, | BC | = | BY |. Show that the XY lines pass through a fixed point for a moving point P.
2004 Slovakia TST p9
A quadrilateral is given such that a sphere centered at point O touches all its six edges.
In addition, four spheres with centers in the vertices of a quadrilateral touching two from the outside and all four touch another sphere centered at point O. Prove that such a quadrilateral must be regular.
2004 Slovakia TST p10
Let ABC be a triangle. On its sides AB, AC lie the points D, E in this order so that
that the line DE is parallel to the line BC. Let P be any interior point of the triangle
ADE and let F, G are successive intersections of the line DE with the lines BP and CP. Let Q be
the second intersection (different from P) of the circles described by the triangles P DG and P F E. Prove that points A, P, Q lie on a straight line.
2004 Slovakia TST p16
Let T be the center of gravity of triangle ABC. Prove that is true
sin | <CAT | + sin | <CBT | <= 2 / \ sqrt 3
2005 Slovakia TST p6
A sharp triangle ABC is given. Let P, N be the heels of its heights from vertices A, B. Nech
K, L are the intersections of the axes of angles BAC, ABC with opposite sides. Let O be the middle described circle and I the center of the inscribed circle of triangle ABC. Prove that the points N, P, I are collinear just when the points L, K, O are collinear.
2005 Slovakia TST p13
The triangle ABC is given. Let us denote successively P, Q, R the heels of the perpendiculars lowered from the vertices A, B, C on the axis of the outer angles of the triangle at the vertices C, A, B. Let d be the diameter of the circle described triangle P QR. Prove that d ^ 2 = p ^ 2 + s ^ 2, where p is the radius of the inscribed circle to the triangle ABC and s is half the perimeter of the triangle ABC.
2006 Slovakia TST p2
Let AA' and BB' be the heights of the acute triangle ABC. Point D lies on the arc of the ACB
the circle described by triangle ABC. Let P be the intersection of the lines AA' and BD, let Q be the intersection of lines BB' and AD. Prove that the line A'B' passes through the center of the line P Q.
2006 Slovakia TST p4
Inside the square with the side of length 6 are located points A, B, C, D such that the distance
any two of these points is at least 5. Prove that points A, B, C, D form a convex
a quadrilateral containing at least 21.
2006 Slovakia TST p12
Let ABCD be a quadrilateral to which applies |<CBD |= 2 |< ADB |, |<ABD | = 2 |< CDB |
a | AB | = | CB |. Prove that | AD | = | CD |.
2007 Slovakia TST p4
Inside BC of the acute triangle ABC lies the point D. The points P and Q are the centers of the circles
described by the triangle ABD and ACD. Prove that there exists a point M different from point A such that it passes all the circles described by all possible triangles AP Q (triangle
ABC is fixed and point D moves along the BC side)
2007 Slovakia TST p11
Let ABC be an isosceles triangle and D be the center of its base BC. Let's mark it below
M is the center AD and N of the heel of height from point D on line BM. Prove that the ANC angle is right.
2007 Slovakia TST p15
Prove that a polygon with a content greater than n can be inserted into a plane by covering
at least n + 1 grid points
2008 Slovakia TST p3
In triangle ABC, denote P by the intersection of the axis of angle BAC with side BC and Q by the intersection axes of angle ABC with side AC. Let M be the intersection of the axis of the angle BAC and the circle described triangle ABC (different from A) and N intersection of the axis of the angle ABC and the circle described by ABC (different from B). Next, on the line AB, choose the points D, E so that D lies on the semi-line opposite to AB and E on a semi-line opposite to BA and at the same time | AD | = | AC | and | BE | = | BC |. Next, let U be the center of the circle of the described triangle BEM and V is the center of the circle of the described triangle ADN.
Let X be the intersection of AU and BV. Show that CX is perpendicular to P Q.
2008 Slovakia TST p10
In the triangle ABC, |] ACB | <|] BAC | <π / 2 and point D lie on the AC side such that
| BD | = | BA |. The circle inscribed in triangle ABC touches side AB at point K and side AC
at point L. Point J is the center of the circle of the inscribed triangle BCD. Prove that the line KL
halves the line AJ
2008 Slovakia TST p12
Let ABCD be a parallelogram, none of its internal angles being 60◦
.Find all pairs of points E and F such that triangles AEB and BF C are isosceles with
bases AB and BC and triangle DEF is equilateral.
2008 Slovakia TST p13
Hunters sit at the tops of a convex polygon with an even number of sides. Inside
of a polygon outside its diagonals is a fox. The hunters suddenly fire towards
fox, but the fox dodges and the bullets from the rifles fly further and fly over the sides of the polygon. Prove that no bullet hits at least one side.
2008 Slovakia TST p18
The triangle ABC is given. The circle assigned to the side BC has the center J and touches
sides BC at point A1 and lines AC, AB respectively at points B1, C1. Suppose that
lines A1B1 and AB are perpendicular to each other and intersect at point D. Let E be the heel of the perpendicular triggered from point C1 on line DJ. Determine the sizes of the angles BEA1 and AEB1.
2009 Slovakia TST p6
The circles k1 and k2 touch on the outside at point K. In addition, they touch on the inside of circle m
at point A1, resp. A2. Let P be one of the intersections of the circle m with the common tangent of the circles k1 and k2 passing through point K. Line P A1 intersects k1 for the second time at point B1, similarly P A2 intersects k2 for the second time at point B2. Prove that B1B2 is a common tangent to the circles k1 and k2
2009 Slovakia TST p11
Let Sa, Sb, Sc be the centers and Ra, Rb, Rc the radii of the circles assigned to the sides BC, CA,
AB triangle ABC. Let sequentially ra, rb, rc denote the radii of the circles inscribed in a triangle
BCSa, ACSb, ABSc. Prove that
r_a / R_a + r_b / R_b + r_c / R_c = 1.
2009 Slovakia TST p15
An isosceles triangle ABC with arms AB and AC is given in the plane. Point M is the center its base BC. Let us choose any point X inside the smaller of the arcs MA on the circle
described triangle ABM. Let T be such a point inside the acute angle BMA that |< TMX | = 90◦
a | T X | = | BX |. Prove that difference |< BTM | - |<MT C | does not depend on the choice of point X
2010 Slovakia TST p2
In the plane there is a given triangle ABC and its described circle k. Circle l with center O
touches the circle k at the point P and the line BC at the point Q. We know that the point P lies on the arc a circle k above the chord BC that does not contain point A. Prove that if |< CAO | =
= |< BAO |, then also |< P AO | = |< QAO |.
2010 Slovakia TST p7
The triangle ABC is inscribed in the circle k. The axes of its internal angles intersect
a circle to the second time at points A0, B0 and C0. Prove that the content of triangle A0B0C0 is greater than or equal to the content of triangle ABC.
2010 Slovakia TST p12
Let a, b, c, d are (in that order) the lengths of the sides AB, BC, CD, DA of the tangent
quadrilateral ABCD. Prove that is true
a ^ 2 / (a + b) + c ^ 2 / (c + d)> = (a + c) / 2
2010 Slovakia TST p14
The convex quadrilateral ABCD is inscribed in a circle. The AB and CD lines are
intersect at point P, lines AD and BC intersect at point Q. Prove that
| P Q | ^ 2 = | P A | · | P B | + | QB | · | QC |
2010 Slovakia TST p17
A trapezoid ABCD with parallel sides AB, CD is given, where | AB | > > | CD |. The points K, L lie successively inside the lines AB, CD such that | AK | / | KB | =
= | DL | / | LC |. The points P, Q lie inside the line KL, where
|< AP B | = |< BCD | a |< CQD | = |< ABC |.
Prove that the points P, Q, B, C lie on one circle.
2011 Slovakia TST p1
Find all finite sets of S points in the plane with the following property: for
every three points A, B, C from the set S there exists a point D from the set S such that the points A, B,
C, D are vertices of the parallelogram
2011 Slovakia TST p3
A sharp isosceles triangle ABC with a base BC is given. For point P
lying inside the triangle ABC, we denote by the successes M and N the intersections of the circle so
center A and radius | AP | with parties AB and AC. Find the point P for which the sum is
| MN | + | BP | + | CP | minimal.
2011 Slovakia TST p4
The string quadrilateral ABCD is given. The semi-lines CB and DA intersect at point P,
the half-lines AB and DC intersect at the point Q. The centers of the diagonals AC and BD are denoted by L and M (respectively). Finally, K be the orthocentre of the triangle MP Q. Prove
that the points P, Q, K, L lie on a circle.
2011 Slovakia TST p10
Let H and O be, respectively, the orthocenter and the center of the described circle to the triangle
ABC. Lines AH and AO intersect the circle at successively at points M and N (different
from A). Let P, Q, R be the intersections of the lines BC and HN, BC and OM, HQ, respectively
and OP. Prove that AORH is a parallelogram.
2011 Slovakia TST p16
A convex pentagon ABCDE is given in which | DC | = | DE | a |< DCB | == |<DEA | = 90◦
. Let F be a point inside the side AB for which | AF | holds : | BF | == | AE | : | BC |. Prove that |<F CE | = |< ADE | a |< F EC | = |<BDC |.
2012 Slovakia TST p1
Let W be the interior point of triangle ABC. The line p1 leads to the point W,
p2, p3 parallel to the sides of the triangle AB, BC and CA which intersect the sides
triangle ABC successively at points K (p1 ∩ CA), N (p1 ∩ BC), L (p2 ∩ AB), O
(p2-CA), M (p3-BC) and P (p3-AB). Diagonals of KB, LC and MA trapezoids
ABNK, BLOC and CMP A divide the triangle ABC into seven parts, four of which are
triangles. Prove that the sum of the contents of three of these triangles that lie at
sides of the triangle ABC, is equal to the contents of the fourth (inner).
2012 Slovakia TST p3
Find the smallest real number k such that: if there is any triangle ABC with pages a <= b <= c, so it exists
a) isosceles triangle XY Z,
b) right isosceles triangle XY Z,
which contains the triangle ABC, and for which the content is S_ {XY Z} <= kb ^ 2
c) How does the result in part b) change if we assume that the triangle ABC is
sharp?
2012 Slovakia TST p5
There are two circles that have an internal contact at the point M and a line that
touches the inner circle at point P and intersects the outer circle at points Q and R.
Prove that the angles QMP and RMP are the same.
2012 Slovakia TST p9
Let ABCDE be a string pentagon. Let a, b, c, d be the successive distances
lines BC, CD, DE and BE from point A. Express d with a, b, c.
2012 Slovakia TST p14
The three different points A, B and C lie on a straight line in this order. Let k be a circle
passing through A and C, the center of which does not lie on the line AC. The tangent to k in points
A and C intersect at point P. The line P B intersects the circle k at point Q. Prove that
the intersection of the angle axis AQC with the line AC is the same regardless of the choice of the circle k.
2012 Slovakia TST p17
Let ABC be an isosceles triangle with base AB. Next, let M be the center AB and P is a point inside the triangle ABC such that |< P AB | = |< P BC |. Prove that |<APM | + |< BP C | = 180◦
.
2013 Slovakia TST p2
The half-lines OA and OB touch the circle k at different points A and B. Let K be internal
point of the shorter arc AB of the circle k. Intersection of the OB semi-straight line with the parallel with the line OA passing through the point K, denote L. The intersection of the line AK with the circle l described by the triangle KLB (different from K) is denoted by M. Prove that the line OM touches the circle l
2013 Slovakia TST p7
Let P, Q and R be points on the sides BC, CA and AB of the acute triangle ABC
such that the triangle P QR is equilateral and has a minimum content among all such
equilateral triangles. Prove that the perpendiculars from points A, B and C respectively to the sides of the QR, RP and P Q intersect at one point.
2013 Slovakia TST p11
Circles k1 and k2 with centers at points O1 and O2 intersect at two points A and B. Lines
O2B and O1B intersect the circles k1 and k2 respectively at points E and F (different from B). Parallelogram with the line EF passing through point B intersects the circles k1 and k2 at points M and N from B). Prove that if the point B lies inside the line MN, then | MN | = | AE | + | AF |.
2014 Slovakia TST p2
A sharp triangle ABC is given. Let B1 be a point on the AC side such that B1B
is the axis of the acute angle ABC. The perpendicular from point B1 to side BC intersects the shorter arc BC the circle described by the triangle ABC at point K. The perpendicular from point B to AK intersects AC at point L. Line B1B intersects the arc AC at point M different from B. Prove that the points K, L, M lie on one line.
2014 Slovakia TST p4
Points A1, B1 and C1 lie in a row inside the sides BC, CA and AB of the triangle ABC.
Let A0, B0 and C0 be the intersections of BB1 ∩ CC1, CC1 ∩ AA1 and AA1 ∩ BB1, respectively.
Prove that if the four triangles CB1A0, AC1B0, BA1C0 and A0B0C0 have the same
content equal to one and three quadrilaterals AB0A0B1, BC0B0C1 and CA0C0A1 have
the same content. Find its size.
2014 Slovakia TST p5
Let α, β, γ be the angles of triangle ABC. Prove that if it applies
(sin α + sin β + sin γ) / (cos α + cos β + cos γ) = \ srt3,
so then one of the angles of the triangle ABC is 60 veľkosť
2014 Slovakia TST p9
On the line AC of the triangle ABC we choose the points M and N so that | AM | = | AB |,
| CN | = | BC | and the points are on a line in the order M, A, C, N. The circles described by the triangle BCM and ABN intersect at points B and K. Prove that BK is the axis of the angle ABC.
2014 Slovakia TST p13
On the sides AD and CD of the parallelogram ABCD with the center S, choose one by one
points P, Q to apply |< ASP | = |< CSQ | = |<ABC |. Prove that
a) the angles ABP and CBQ are identical,
b) the lines AQ and CP intersect at the circle of the described triangle ABC.
2015 Slovakia TST p4
On the side BC of the triangle ABC lies the point M such that the center of gravity of the triangle ABM lies on the circle of the described triangle ACM and at the same time the center of gravity of the triangle ACM lies on the circle of the described triangle ABM. Prove that the line of the triangles ABM and ACM from point M are equally long
2015 Slovakia TST p8.
Let us have a right triangle ABC with a right angle at the vertex B. Let BD be
height from the top B to the AC side (point D lies on the AC). Next, let us denote P, Q and I respectively the centers of the circles inscribed in the triangles ABD, CBD and ABC. Show that the center of the circle of the described triangle P IQ lies on the line AC
2015 Slovakia TST p15
Let ABC be a triangle with the center of the described circle at the point O and let D be
the intersection of the axis of the angle BAC and the line BC. Furthermore, let M be such a point that MC ⊥ BC and MA ⊥ AD and let the lines BM and OA intersect at the point P. Prove that the line BC
is a tangent to a circle centered on P and passing through A
2016 Slovakia TST p2
In the acute triangle ABC, point D lies on the side BC. Let the points O1, O2 denote
the centers of the circles described by the triangle ABD and ACD. Prove that the line joining the center
described circle ABC and the orthocenter of triangle O1O2D is parallel to BC.
2016 Slovakia TST p5
Let ABCD be a string quadrilateral inscribed in a circle with center O. Let E, F be
successively the centers of the arcs AB, CD, which do not contain the remaining vertices of the quadrilateral. Bodmi E, F let us make lines parallel to the diagonals of the quadrilateral ABCD. These four lines are intersect at points E, F, K, L. Prove that the points K, O, L lie on one line
2016 Slovakia TST p9
Given an acute triangle ABC, the intersection of the heights is denoted H. Let D be such
the point that the quadrilateral HABD is a parallelogram (i.e. AB to HD and AH to BD). Let's denote E ten a point lying on the line DH for which the line AC passes through the center of the line HE. Point F is another intersection of the line AC with the circle described by the triangle DCE. Prove that | EF | = | AH |.
2016 Slovakia TST p15
In the string quadrilateral ABCD, the tangent to its described circle at points A, C
intersect on the line BD. Let M be the center of AC. The parallel with BC through D intersects the line BM at point E and the circle described by ABCD at point F, F \ne D. Prove that BCEF is a parallelogram.
2016 Slovakia TST p16
Let I be the center of the inscribed circle of the triangle ABC, where | AB | \ne | AC |. Let us denote M center of side BC and D point of contact of the inscribed circle with side BC. Circle with center
at point M and radius MD intersects the line AI at points P and Q. Prove that |<BAC | + |< PMQ | = 180◦
2017 Slovakia TST p2
Prove that in each triangle with sides a, b, c and the radius of the circle R described
applies 9R ^ 2> = a ^ 2 + b ^ 2 + c ^ 2
.
2017 Slovakia TST p3
Let ABC be a triangle with an inscribed circle k. Let's mark the points of contact with the sides BC
and AC sequentially D1 and E1. Points D2 and E2 lie on the sides BC and AC, with | CD2 | = | BD1 | and | CE2 | = | AE1 |. The intersection of AD2 and BE2 is denoted by P. Circle to intersect AD2 at two points, of which the one closer to the vertex A is denoted by Q. Prove that | AQ | = | D2P |.
2017 Slovakia TST p10
The triangle ABC is given. A line parallel to the BC side intersects the AB and AC sides
successively at points P and Q. Let M be the interior point of triangle AP Q. Lines MB and MC
intersect the line P Q successively at points E and F. Let N be the second intersection of the circles described triangle PMF and QME. Prove that the points A, M, N lie on one line.
2017 Slovakia TST p12
Let ABCD be a chord quadrilateral with the described circle k and let r and s be successive
images of the line AB in axial symmetry according to the axis of the internal angle CAD and CBD. Straight lines r and s intersect at a point P located in the outer region of the circle k. Let's mark O middle circle k. Prove that the lines OP and CD are perpendicular to each other.
2017 Slovakia TST p16
A scalene acute triangle ABC is given. Let H be its orthocenter and O the middle
circle described. Furthermore, let B0 = BH ∩AC, C0 = CH ∩AB, P = AH ∩B0C0 and T = AO∩BC.
Let M be the center of the side BC. Prove that MH // T P
source: https://skmo.sk/
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