geometry problems from China Western Mathematical Olympiads / Invitation (CWMI)

with aops links in the names

with aops links in the names

2001 - 2019

ABCD is a rectangle of area 2. P is a point on side CD and Q is the
point where the incircle of △PAB touches the
side AB. The product PA・PB varies as ABCD and P vary. When PA·PB attains its minimum value.

a) Prove that AB ≥ 2BC,

b) Find the value of AQ · BQ.

P is a point on the exterior of a circle centered at O. The tangents to
the circle from P touch the circle at A and B. Let Q be the point of
intersection of PO and AB. Let CD be any chord of the circle passing through Q.
Prove that △PAB and △PCD have the
same incentre.

Let O be the circumcenter of acute triangle ABC. Point P is in the
interior of triangle AOB. Let D,E, F be the projections of P on the sides BC, CA,
AB, respectively. Prove that the parallelogram consisting of FE and FD as its adjacent
sides lies inside triangle ABC.

Given a trapezoid ABCD with AD // BC, E is a moving point on
the side AB, let O

_{1},O_{2}be the circumcenters of triangles AED, BEC, respectively. Prove that the length of O_{1}O_{2}is a constant value.
Given that the sum of the distances from point P in the interior of a
convex quadrilateral ABCD to the sides AB, BC, CD, DA is a constant, prove that
ABCD is a parallelogram.

A circle can be inscribed in the convex quadrilateral ABCD. The incircle
touches the sides AB,BC,CD,DA at A

_{1},B_{1},C_{1},D_{1}respectively. The points E,F,G,H are the midpoints of A_{1}B_{1},B_{1}C_{1},C_{1}D_{1}, D_{1}A_{1}respectively. Prove that the quadrilateral EFGH is a rectangle if and only if A,B,C,D are concyclic.
Let ABCD be a convex quadrilateral, I

_{1}and I_{2}be the incenters of triangles ABC and DBC respectively. The line I_{1}I_{2}intersects the lines AB and DC at points E and F respectively. Given that AB and CD intersect in P, and PE = PF, prove that the points A,B,C,D lie on a circle.
Let ℓ be the perimeter of an acute-angled triangle ABC which is not an
equilateral triangle. Let P be a variable points inside the triangle ABC, and
let D,E, F be the projections of P on the sides BC,CA,AB respectively. Prove that
2(AF + BD + CE) = ℓ if and only if P is collinear with the incenter and the
circumcenter of the triangle ABC.

Given three points P, A, B and a circle such that the lines PA and PB
are tangent to the circle at the points A and B, respectively. A line through
the point P intersects that circle at two points C and D. Through the point B, draw
a line parallel to PA, let this line intersect the lines AC and AD at the points
E and F, respectively. Prove that BE = BF.

Circles C(O

_{1}) and C(O_{2}) intersect at points A, B. CD passing through point O_{1}intersects C(O_{1}) at point D and tangents C(O_{2}) at point C. AC tangents C(O_{1}) at A. Draw AE $\perp$ CD, and AE intersects C(O_{1}) at E. Draw AF $\perp$ DE, and AF intersects DE at F. Prove that BD bisects AF.
In isosceles right-angled triangle ABC, CA = CB = 1. P is an arbitrary
point on the sides of ABC. Find the maximum of PA·PB ·PC.

In △PBC, ÐPBC = 60

^{o}, the tangent at point P to the circumcircleg of △PBC intersects with line CB at A. Points D and E lie on the line segment PA and g respectively, satisfying ÐDBE = 90^{o}, PD = PE. BE and PC meet at F. It is known that lines AF,BP,CD are concurrent.
a) Prove that BF bisect ÐPBC

b) Find tan ÐPCB

AB is a diameter of the circle O, the point C lies on the line AB
produced. A line passing though C intersects with the circle O at the point D
and E. OF is a diameter of circumcircle O1 of △BOD. Join CF and produce, cutting the circle O1 at G. Prove that points
O, A,E,G are concyclic.

Let C and D be two intersection points of circle O1 and circle O2. A
line, passing through D, intersects the circle O1 and the circle O2 at the
points A and B respectively. The points P and Q are on circles O1 and O2
respectively. The lines PD and AC intersect at H, and the lines QD and BC
intersect at M. Suppose that O is the circumcenter of the triangle ABC. Prove
that

OD ⊥ MH if and only if P,Q,M and H are concyclic.

Let P be an interior point of an acute angled triangle ABC. The lines
AP,BP,CP meet BC,CA,AB at points D,E, F respectively. Given that triangle △DEF and △ABC are similar, prove that P is the centroid of △ABC.

In triangle ABC, AB = AC, the inscribed circle I touches BC,CA,AB at points
D,E and F respectively. P is a point on arc EF opposite D. Line BP intersects
circle I at another point Q, lines EP, EQ meet line BC at M,N respectively.
Prove that

a) P, F, B, M concyclic

b) EM / EN = BD / BP .

Let H be the orthocenter of acute triangle ABC and D the midpoint of BC.
A line through H intersects AB,AC at F,E respectively, such that AE = AF. The
ray DH intersects the circumcircle of △ABC at P. Prove
that P, A,E, F are concyclic.

Given an acute triangle ABC, D is a point on BC. A circle with diameter BD
intersects line AB,AD at X, P respectively (different from B,D).The circle with
diameter CD intersects AC,AD at Y,Q respectively (different from C,D). Draw two
lines through A perpendicular to PX,QY , the feet are M,N respectively. Prove
that △AMN is similar to △ABC if and only
if AD passes through the circumcenter of △ABC.

AB is a diameter of a circle with center O. Let C and D be two different
points on the circle on the same side of AB, and the lines tangent to the
circle at points C and D meet at E. Segments AD and BC meet at F. Lines EF and AB
meet at M. Prove that E,C,M and D are concyclic.

△ABC is a right-angled triangle, ÐC = 90

^{o}. Draw a circle centered at B with radius BC. Let D be a point on the side AC, and DE is tangent to the circle at E. The line through C perpendicular to AB meets line BE at F. Line AF meets DE at point G. The line through A parallel to BG meets DE at H. Prove that GE = GH.
In a circle Γ

_{1}, centered at O, AB and CD are two unequal in length chords intersecting at E inside Γ_{1}. A circle Γ_{2}, centered at I is tangent to Γ_{1}internally at F, and also tangent to AB at G and CD at H. A line*l*through O intersects AB and CD at P and Q respectively such that EP=EQ. The line EF intersects*l*at M. Prove that the line through M parallel to AB is tangent to Γ_{1}.
In triangle ABC with AB > AC and incenter I, the incircle touches
BC,CA,AB at D,E, F respectively. M is the midpoint of BC, and the altitude at A
meets BC at H. Ray AI meets lines DE and DF at K and L, respectively. Prove that
the points M,L,H,K are concyclic.

P is a point in the _ABC, ω is the
circumcircle of _ABC. BP Ç ω ={B,B

_{1}},CP Ç ω = {C,C_{1}}, PE $\perp$ AC, PF $\perp$ AB. The radius of the inscribed circle and circumcircle of △ABC is r, R. Prove EF / B_{1}C_{1}≥ r / R .
O is the circumcenter of acute △ABC, H is the orthocenter.
AD $\perp$ BC, EF is the perpendicular bisector of AO, D,E on the BC. Prove that
the circumcircle of △ADE through the midpoint of OH.

Let ABC be a triangle, and B1,C1 be its excenters opposite B,C. B

_{2},C_{2}are reflections of B_{1},C_{1}across midpoints of AC,AB. Let D be the extouch at BC. Show that AD is perpendicular to B_{2}C_{2}.
Let PA, PB be tangents to a circle centered at O, and C a point on the
minor arc AB. The perpendicular from C to PC intersects internal angle
bisectors of AOC,BOC at D,E. Show that CD = CE.

Let AB be the diameter of semicircle O , C, D be points on the arc AB,
P,Q be respectively the circumcenter of △OAC and △OBD . Prove that: CP·CQ = DP·DQ.

In the plane, Point O is the center of the equilateral triangle ABC ,
Points P,Q such that $\overrightarrow{OQ}=2\overrightarrow{PO}$. Prove that |PA| + |PB| + |PC| ≤ |QA| + |QB| + |QC|.

Two circles (Ω

_{1}) , (Ω_{2}) touch internally on the point T. Let M,N be two points on the smaller circle (Ω_{1}) which are different from T and A,B,C,D be four points on (Ω_{2}) such that the chords AB,CD pass through M,N, respectively. Prove that if AC,BD,MN have a common point K, then TK is the angle bisector of ÐMTN.
Let a, b, c, d are lengths of the sides of a convex quadrangle with the
area equal to S, set S={x

_{1}, x_{2}, x_{3}, x_{4}} consists of permutations x_{i}of (a, b, c, d). Prove that S ≤ 1/2 (x_{1}x_{2}+ x_{3}x_{4}).
Let circle (O

_{1}) and circle (O_{2}) intersect at P and Q, their common external tangent touches (O_{1}) and (O_{2}) at A and B respectively. A circle Γ passing through A and B intersects (O_{1}), (O_{2}) at D, C. Prove that CP / CQ= DP / DQ.
ABCD is a cyclic quadrilateral, and ÐBAC = ÐDAC. Points I

_{1}and I_{2}are the incircles of △ABD and △ADC respectively. Prove that one of the common external tangents of incircles (I_{1}) and (I_{2}) is parallel to BD
In triangle ABC, let D be a point on BC. Let I

_{1}and I_{2}be the incenters of triangles ABD and ACD respectively. Let O_{1}and O_{2}be the circumcenters of triangles AI_{1}D and AI_{2}D respectively. Let lines I_{1}O_{2}and I_{2}O_{1}meet at P. Show that PD $\perp$ BC.
In acute triangle ABC, let D and E be points on sides AB and AC
respectively. Let segments BE and DC meet at point H. Let M and N be the
midpoints of segments BD and CE respectively. Show that H is the orthocenter of
triangle AMN if and only if B,C,E,D are concyclic and BE $\perp$ CD.

In acute angled $\triangle ABC$, $AB > AC$, points $E, F$ lie on $AC, AB$ respectively, satisfying $BF+CE = BC$. Let $I_B, I_C$ be the excenters of $\triangle ABC$ opposite $B, C$ respectively, $EI_C, FI_B$ intersect at $T$, and let $K$ be the midpoint of arc $BAC$. Let $KT$ intersect the circumcircle of $\triangle ABC$ at $K,P$. Show $T,F,P,E$ concyclic.

In acute triangle $ABC,$ $AB<AC,$ $O$ is the circumcenter of the triangle. $M$ is the midpoint of segment $BC,$ $(AOM)$ intersects the line $AB$ again at $D$ and intersects the segment $AC$ at $E.$ Prove that $DM=EC.$

Let $O,H$ be the circumcenter and orthocenter of acute triangle $ABC$ with $AB\neq AC$, respectively. Let $M$ be the midpoint of $BC$ and $K$ be the intersection of $AM$ and the circumcircle of $\triangle BHC$, such that $M$ lies between $A$ and $K$. Let $N$ be the intersection of $HK$ and $BC$. Show that if $\angle BAM=\angle CAN$, then $AN\perp OH$.

In acute-angled triangle $ABC,$ $AB>AC.$ Let $O,H$ be the circumcenter and orthocenter of $\triangle ABC,$ respectively. The line passing through $H$ and parallel to $AB$ intersects line $AC$ at $M,$ and the line passing through $H$ and parallel to $AC$ intersects line $AB$ at $N.$ $L$ is the reflection of the point $H$ in $MN.$ Line $OL$ and $AH$ intersect at $K.$ Prove that $K,M,L,N$ are concyclic.

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