### China Western 2001-19 (CWMI) 38p

geometry problems from China Western Mathematical Olympiads / Invitation (CWMI)
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 PAPB 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 O1,O2 be the circumcenters of triangles AED, BEC, respectively. Prove that the length of O1O2 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 A1,B1,C1,D1 respectively. The points E,F,G,H are the midpoints of A1B1,B1C1,C1D1, D1A1 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, I1 and I2 be the incenters of triangles ABC and DBC respectively. The line I1I2 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(O1) and C(O2) intersect at points A, B. CD passing through point O1 intersects C(O1) at point D and tangents C(O2) at point C. AC tangents C(O1) at A. Draw AE $\perp$ CD, and AE intersects C(O1) 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 = 60o, 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 = 90o, 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 = 90o. 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 Γ1, centered at O, AB and CD are two unequal in length chords intersecting at E inside Γ1. A circle Γ2, centered at I is tangent to Γ1 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 Γ1.

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,B1},CP Ç ω = {C,C1}, PE $\perp$ AC, PF $\perp$ AB. The radius of the inscribed circle and circumcircle of ABC is r, R. Prove  EF / B1C1 ≥ 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. B2,C2 are reflections of B1,C1 across midpoints of AC,AB. Let D be the extouch at BC. Show that AD is perpendicular to B2C2.

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 (Ω1) , (Ω2) touch internally on the point T. Let M,N be two points on the smaller circle (Ω1) which are different from T and A,B,C,D be four points on (Ω2) 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={x1, x2, x3, x4} consists of permutations xi of (a, b, c, d). Prove that S 1/2 (x1x2 + x3x4).

Let circle (O1) and circle (O2) intersect at P and Q, their common external tangent touches (O1) and (O2) at A and B respectively. A circle Γ passing through A and B intersects  (O1), (O2) at D, C. Prove that   CP / CQ= DP / DQ.

ABCD is a cyclic quadrilateral, and ÐBAC = ÐDAC. Points I1 and I2 are the incircles of ABD and ADC respectively. Prove that one of the common external tangents of  incircles (I1) and (I2) is parallel to BD

In triangle ABC, let D be a point on BC. Let I1 and I2 be the incenters of triangles ABD and ACD respectively. Let O1 and O2 be the circumcenters of triangles AI1D and AI2D respectively. Let lines I1O2 and I2O1 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.