Here are collected

with aops links (no contest's problems)

**all**the Euclidean Geometry**problems (**with or without aops links) from the problem corner (only those without any constest's source) and the geometry**articles**from the online magazine ''Mathematical Excalibur''.__geometry__

__problems from problem corner__

with aops links (no contest's problems)

collected inside aops

**here**
P4 If the diagonals of a quadrilateral in the plane are perpendicular. show that the midpoints of its sides and the feet of the perpendiculars dropped from the midpoints to the opposite sides lie on a circle.

P9 On sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ with $AB < CD$, locate points $F$ and $E$, respectively, such that $\frac{AF}{FD}=\frac{BE}{EC} =\frac{AB}{CD}$. Suppose $EF$ when extended beyond $F$ meets line $BA$ at $P$ and meets line $CD$ at $Q$. Show that $\angle BPE = \angle CQE$.

P14 Suppose $\triangle ABC, \triangle A'B'C'$ are (directly) similar to each other and $\triangle AA'A'', \triangle BB'B''. \triangle CC'C'' $ are also (directly) similar to each other. Show that \triangle $A''B''C''$ is (directly) similar to $\triangle ABC$.

P19 Suppose $A$ is a point inside a given circleand is different from the canter. Consider all chords (excludinh the diameter) passing through $A$. What is the locus of intersections of the tangent lines at the endpoints of these chords?

P27 Let $ABCD$ be a cyclic quadrilateral and let $I_A, I_B, I_C, I_D$ be the incenters of $\triangle BCD, \triangle ACD, \triangle ABD, \triangle ABC$, respectively. Show that $I_AI_BI_CI_D$ is a rectangle.

P48 Squares $ABDE$ and $BCFG$ are drawn outside of triangle $ABC$. Prove that triangle $ABC$ is isosceles if DG is parallel to $AC$.

P53 For $\triangle ABC$, define $A'$ on $BC$ so that $AB + BA' = AC + CA'$ and similarly define $B'$ on $CA$ and $C'$ on $AB$. Show that $AA', BB', CC'$ are concurrent.

P84 Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$ of $\triangle ABC$, respectively. Draw an arbitrary line through $A$. Let $Q$ and $R$ be the feet of the perpendiculars from $B$ and $C$ to this line, respectively. Find the locus of the intersection $P$ of the lines $QM$ and $RN$ as the line rotates about $A$.

P115 Find the locus of the points $P$ in the plane of an equilateral triangle $ABC$ for which the triangle formed with lengths $PA, PB$ and $PC$ has constant area.

P158 Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D$ be a point on $B$C such that $BD = 2DC$ and let $P$ be a point on $AD$ such that $\angle BAC = \angle BPD$. Prove that $\angle BAC = 2 \angle DPC$.

P164 Let $O$ be the center of the excircle of triangle $ABC$ opposite $A$. Let $M$ be the midpoint of $AC$ and let $P$ be the intersection of lines $MO$ and $BC$. Prove that if $\angle BAC = 2 \angle ACB$, then $AB = BP$.

P168 Let $AB$ and $CD$ be nonintersecting chords of a circle and let $K$ be a point on $CD$. Construct (with straightedge and compass) a point $P$ on the circle such that $K$ is the midpoint of the part of segment $CD$ lying inside triangle $ABP$.

P174 Let $M$ be a point inside acute triangle $ABC$. Let $A', B', C'$ be the mirror images of $M$ with respect to $BC, CA, AB$, respectively. Determine (with proof) all points $M$ such that $A, B, C, A', B', C'$ are concyclic.

P179 Prove that in any triangle, a line passing through the incenter cuts the perimeter of the triangle in half if and only if it cuts the area of the triangle in half .

P184 Let $ABCD$ be a rhombus with $\angle B = 60^o$ . $M$ is a point inside $\triangle ADC$ such that $\angle AMC = 120^o$ . Let lines $BA$ and $CM$ intersect at $P$ and lines $BC$ and $AM$ intersect at $Q$. Prove that $D$ lies on the line $PQ$.

P188 The line $S$ is tangent to the circumcircle of acute triangle $ABC$ at $B$. Let $K$ be the projection of the orthocenter of triangle $ABC$ onto line $S$. Let $L$ be the midpoint of side $AC$. Show that triangle $BKL$ is isosceles

P194 A circle with center $O$ is internally tangent to two circles inside it, with centers $O_1$ and $O_2$, at points $S$ and $T$ respectively. Suppose the two circles inside intersect at points $M, N$ with $N$ closer to $ST$. Show that $S, N, T$ are collinear if and only if $\frac{SO_1}{OO_1} = \frac{OO_2}{TO_2}$.

P198 In a triangle $ABC, AC = BC$. Given is a point $P$ on side $AB$ such that $\angle ACP = 30^o$. In addition, point $Q$ outside the triangle satisfies $\angle CPQ=\angle CPA + \angle APQ = 78^o$. Given that all angles of triangles $ABC$ and $QPB$, measured in degrees, are integers, determine the angles of these two triangles.

P202 For triangle $ABC$, let $D, E, F$ be the midpoints of sides $AB, BC, CA$, respectively. Determine which triangles $ABC$ have the property that triangles $ADF, BED, CFE$ can be folded above the plane of triangle $DEF$ to form a tetrahedron with $AD$ coincides with $BD,BE$ coincides with $CE, CF$ coincides with $AF$.

P208 In $\triangle ABC, AB > AC > BC$.Let $D$ be a point on the minor arc $BC$ of the circumcircle of $\triangle ABC$. Let $O$ be the circumcenter of $\triangle ABC$. Let $E, F$ be the intersection points of line $AD$ with the perpendiculars from $O$ to $AB, AC$, respectively. Let $P$ be the intersection of lines $BE$ and $CF$. If $PB = PC + PO$, then find $\angle BAC$ with proof.

P214 Let the inscribed circle of triangle $ABC$ be tangent to sides $AB, BC$ at $E$ and $F$ respectively. Let the angle bisector of $\angle CAB$ intersect segment $EF$ at $K$. Prove that $\angle CKA$ is a right angle.

P228 In $\triangle ABC, M$ is the foot of the perpendicular from $A$ to the angle bisector of $\angle BCA$. $N$ and $L$ are respectively the feet of perpendiculars from $A$ and $C$ to the bisector of $\angle ABC$. Let $F$ be the intersection of lines $MN$ and $AC$. Let $E$ be the intersection of lines $BF$ and $CL$. Let $D$ be the intersection of lines $BL$ and $AC$. Prove that lines $DE$ and $MN$ are parallel.

P245 $ABCD$ is a concave quadrilateral such that $\angle BAD =\angle ABC =\angle CDA = 45^o$. Prove that $AC = BD$.

P248 Let $ABCD$ be a convex quadrilateral such that line $CD$ is tangent to the circle with side $AB$ as diameter. Prove that line $AB$ is tangent to the circle with side $CD$ as diameter if and only if lines $BC$ and $AD$ are parallel.

P253 Suppose the bisector of $\angle BAC$ intersect the arc opposite the angle on the circumcircle of $\triangle ABC$ at $A_1$. Let $B_1$ and $C_1$ be defined similarly. Prove that the area of $\triangle A_1B_1C_1$ is at least the area of $\triangle ABC$.

P262 Let $O$ be the center of the circumcircle of $\triangle ABC$ and let $AD$ be a diameter. Let the tangent at $D$ to the circumcircle intersect line $BC$ at $P$. Let line $PO$ intersect lines $AC, AB$ at $M, N$ respectively. Prove that $OM = ON$.

P268, 337 In triangle $ABC, \angle ABC = \angle ACB = 40^o$. Points $P$ and $Q$ are inside the triangle such that $\angle PAB = \angle QAC = 20^o$ and $\angle PCB = \angle QCA = 10^o$ . Must $B, P, Q$ be collinear? Give a proof.

P272 $\triangle ABC$ is equilateral. Find the locus of all point $Q$ inside the triangle such that $ \angle QAB + \angle QBC + \angle QCA =90^o$ .

P278 Line segment $SA$ is perpendicular to the plane of the square $ABCD$. Let $E$ be the foot of the perpendicular from $A$ to line segment $SB$. Let $P, Q, R$ be the midpoints of $SD, BD, CD$ respectively. Let $M, N$ be on line segments $PQ, PR$ respectively. Prove that $AE$ is perpendicular to $MN$.

P288 (also) Let $H$ be the orthocenter of triangle $ABC$. Let $P$ be a point in the plane of the triangle such that $P$ is different from $A, B, C$. Let $L, M, N$ be the feet of the perpendiculars from $H$ to lines $PA, PB, PC$ respectively. Let $X, Y, Z$ be the intersection points of lines $LH, MH, NH$ with lines $BC, CA, AB$ respectively. Prove that $X, Y, Z$ are on a line perpendicular to line $PH$.

P293 Let $CH$ be the altitude of triangle $ABC$ with $\angle ACB = 90^o$. The bisector of $\angle BAC$ intersects $CH, CB$ at $P, M$ respectively. The bisector of $\angle ABC$ intersects $CH, CA$ at $Q, N$ respectively. Prove that the line passing through the midpoints of $PM$ and $QN$ is parallel to line $AB$.

P298 The diagonals of a convex quadrilateral $ABCD$ intersect at $O$. Let $M_1$ and $M_2$ be the centroids of $\triangle AOB$ and $\triangle COD$ respectively. Let $H_1$ and $H_2$ be the orthocenters of $\triangle BOC$ and $\triangle DOA$ respectively. Prove that $M_1M_2 \perp H_1H_2$.

305 A circle $\Gamma_2$ is internally tangent to the circumcircle $\Gamma_1$ of $\triangle PAB$ at $P$ and side $AB$ at $C$. Let $E, F$ be the intersection of $\Gamma_2$ with sides $PA, PB$ respectively. Let $EF$ intersect $PC$ at $D$. Lines $PD, AD$ intersect $\Gamma_1$ again at $G, H$ respectively. Prove that $F, G, H$ are collinear.

309 (ISL 2002 G7) In acute triangle $ABC, AB > AC$. Let $H$ be the foot of the perpendicular from $A$ to $BC$ and $M$ be the midpoint of $AH$. Let $D$ be the point where the incircle of $\triangle ABC$ is tangent to side $BC$. Let line $DM$ intersect the incircle again at $N$. Prove that $\angle BND = \angle CND$.

313 In $\triangle ABC, AB < AC$ and $O$ is its circumcenter. Let the tangent at $A$ to the circumcircle cut line $BC$ at $D$. Let the perpendicular lines to line $BC$ at $B$ and $C$ cut the perpendicular bisectors of sides $AB$ and $AC$ at $E$ and $F$ respectively. Prove that $D, E, F$ are collinear.

321 Let $AA', BB'$ and $CC'$ be three non-coplanar chords of a sphere and let them all pass through a common point P inside the sphere. There is a (unique) sphere $S_1$ passing through $A, B, C, P$ and a (unique) sphere $S_2$ passing through $A', B', C', P$. If $S_1$ and $S_2$ are externally tangent at $P$, then prove that $AA'=BB'=CC'$.

324 $ADPE$ is a convex quadrilateral such that $\angle ADP = \angle AEP$. Extend side $AD$ beyond $D$ to a point $B$ and extend side $AE$ beyond $E$ to a point $C$ so that $\angle DPB = \angle EPC$. Let $O_1$ be the circumcenter of $\triangle ADE$ and let $O_2$ be the circumcenter of $\triangle ABC$. If the circumcircles of $\triangle ADE$ and $\triangle ABC$ are not tangent to each other, then prove that line $O_1O_2$ bisects line segment $AP$.

332 Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. Let $BD$ bisect $OC$ perpendicularly. On diagonal $AC$, choose the point $P$ such that $PC=OC$. Let line $BP$ intersect line $AD$ and the circumcircle of $ABCD$ at $E$ and $F$ respectively. Prove that $PF$ is the geometric mean of $EF$ and $BF$ in length.

344 (Bulgaria 97) $ABCD$ is a cyclic quadrilateral. Let $M, N$ be midpoints of diagonals $AC, BD$ respectively. Lines $BA, CD$ intersect at $E$ and lines $AD, BC$ intersect at $F$. Prove that $ \big| \frac{BD}{AC}-\frac{AC}{BD}\big|= \frac{2MN}{EF}$

without aops links (no contest's problems)

P355 In a plane, there are two similar convex quadrilaterals $ABCD$ and $AB_1C_1D_1$ such that $C, D$ are inside $AB_1C_1D_1$ and $B$ is outside $AB_1C_1D_1$. Prove that if lines $BB_1, CC_1$ and $DD_1$ concur, then $ABCD$ is cyclic. Is the converse also true?

P358 $ABCD$ is a cyclic quadrilateral with $AC$ intersects $BD$ at $P$. Let $E, F, G, H$ be the feet of perpendiculars from $P$ to sides $AB, BC, CD, DA$ respectively. Prove that lines $EH, BD, FG$ are concurrent or are parallel.

P363 Extend side $CB$ of triangle $ABC$ beyond $B$ to a point $D$ such that $DB=AB$. Let $M$ be the midpoint of side $AC$. Let the bisector of $\angle ABC$ intersect line $DM$ at $P$. Prove that $\angle BAP =\angle ACB$.

P369 $ABC$ is a triangle with $BC > CA > AB$. $D$ is a point on side $BC$ and $E$ is a point on ray $BA$ beyond $A$ so that $BD=BE=CA$. Let $P$ be a point on side $AC$ such that $E, B, D, P$ are concyclic. Let $Q$ be the intersection point of ray $BP$ and the circumcircle of $\triangle ABC$ different from $B$. Prove that $AQ+CQ=BP$ .

P374 $O$ is the circumcenter of acute $\triangle ABC$ and $T$ is the circumcenter of $\triangle AOC$. Let $M$ be the midpoint of side $AC$. On sides $AB$ and $BC$, there are points $D$ and $E$ respectively such that $\angle BDM=\angle BEM=\angle ABC$. Prove that $BT\perp DE$.

P379 Let $\ell$ be a line on the plane of $\triangle ABC$ such that $\ell$ does not intersect the triangle and none of the lines $AB, BC, CA$ is perpendicular to $\ell$. Let $A', B', C'$ be the feet of the perpendiculars from $A, B, C$ to $\ell$ respectively. Let $A'', B'', C''$ be the feet of the perpendiculars from $A', B',C'$ to lines $BC, CA, AB$ respectively. Prove that lines $A'A'', B'B'', C'C''$ are concurrent.

P383 Let $O$ and $I$ be the circumcenter and incenter of $\triangle ABC$ respectively. If $AB\ne AC$, points $D, E$ are midpoints of $AB, AC$ respectively and $BC=(AB+AC)/2$, then prove that the line $OI$ and the bisector of $\angle CAB$ are perpendicular .

P388 In $\triangle ABC, \angle BAC=30^o$ and $\angle ABC=70^o$. There is a point $M$ lying inside $\triangle ABC$ such that $\angle MAB= \angle MCA=20^o$. Determine $\angle MBA$ (with proof).

P394 Let $O$ and $H$ be the circumcenter and orthocenter of acute $\triangle ABC$. The bisector of $\angle BAC$ meets the circumcircle $\Gamma$ of $\triangle ABC$ at $D$. Let $E$ be the mirror image of $D$ with respect to line $BC$. Let $F$ be on $\Gamma$ such that $DF$ is a diameter. Let lines $AE$ and $FH$ meet at $G$. Let $M$ be the midpoint of side $BC$. Prove that $GM\perp A$F.

P399 Let $ABC$ be a triangle for which $\angle BAC=60^o$. Let P be the point of intersection of the bisector of $\angle ABC$ and the side $AC$. Let $Q$ be the point of intersection of the bisector of $\angle ACB$ and the side $AB$. Let $r_1$ and $r_2$ be the radii of the incircles of triangles $ABC$ and $APQ$ respectively. Determine the radius of the circumcircle of triangle $APQ$ in terms of $r_1$ and $r_2$ with proof.

P404 Let $I$ be the incenter of acute $\triangle ABC$. Let $\Gamma$ be a circle with center $I$ that lies inside $\triangle ABC$. $D, E, F$ are the intersection points of circle $\Gamma$ with the perpendicular rays from $I$ to sides $BC, CA, AB$ respectively. Prove that lines $AD, BE, CF$ are concurrent.

P407 Three circles $S, S_1, S_2$ are given in a plane. $S_1$ and $S_2$ touch each other externally, and both of them touch $S$ internally at $A_1$ and at $A_2$ respectively. Let $P$ be one of the two points where the common internal tangent to $S_1$ and $S_2$ meets $S$. Let $B_i$ be the intersection points of $PA_i$ and $S_i$ ($i=1,2$). Prove that line $B_1B_2$ is a common tangent to $S_1$ and $S_2$.

P412 $\triangle ABC$ is equilateral and points $D, E, F$ are on sides $BC, CA, AB$ respectively. If $\angle BAD +\angle CBE + \angle ACF = 120^o$,

then prove that $\triangle BAD, \triangle CBE$ and $\triangle ACF$ cover $\triangle ABC$.

P415 [also]Given a triangle $ABC$ such that $\angle BAC=103^o$ and $\angle ABC=51^o$. Let $M$ be a point inside ΔABC such that $\angle MAC=30^o$ and $\angle MCA=13^o$. Find $\angle MBC$ with proof, without trigonometry.

P418 Point $M$ is the midpoint of side $AB$ of acute $\triangle ABC$. Points $P$ and $Q$ are the feet of perpendicular from $A$ to side $BC$ and from $B$ to side $AC$ respectively. Line $AC$ is tangent to the circumcircle of $\triangle BMP$. Prove that line $BC$ is tangent to the circumcircle of $\triangle AMQ$.

P421 For every acute triangle $ABC$, prove that there exists a point $P$ inside the circumcircle $\omega$ of ∆ABC such that if rays $AP, BP, CP$ intersect $\omega$ at $D, E, F$, then $DE: EF: FD = 4:5:6$.

P428 Let $A_1A_2A_3A_4$ be a convex quadrilateral. Prove that the nine point circles of $\triangle A_1A_2A_3, \triangle A_2A_3A_4, \triangle A_3A_4A_1$ and $\triangle A_4A_1A_2$ pass through a common point.

P434 Let $O$ and $H$ be the circumcenter and orthocenter of $\triangle ABC$ respectively. Let $D$ be the foot of perpendicular from $C$ to side $AB$. Let $E$ be a point on line $BC$ such that $ED \perp OD$. If the circumcircle of $\triangle BCH$ intersects side $AB$ at $F$, then prove that points $E, F, H$ are collinear.

P439 In acute triangle $ABC, T$ is a point on the altitude $AD$ (with $D$ on side $BC$). Lines $BT$ and $AC$ intersect at $E$, lines $CT$ and $AB$ intersect at $F$, lines $EF$ and $AD$ intersect at $G$. A line $\ell$ passing through $G$ intersects side $AB$, side $AC$, line $BT$, line $CT$ at $M, N, P, Q$ respectively. Prove that $\angle MDQ =\angle NDP$.

P444 Let $D$ be on side $BC$ of equilateral triangle $ABC$. Let $P$ and $Q$ be the incenters of $\triangle ABD$ and $\triangle ACD$ respectively. Let $E$ be the point so that $\triangle EPQ$ is equilateral and $D, E$ are on opposite sides of line $PQ$. Prove that lines $BC$ and $DE$ are perpendicular.

P450 Let $A_1A_2A_3$ be a triangle with no right angle and $O$ be its circumcenter. For $i = 1,2,3$, let the reflection of $A_i$ with respect to $O$ be $A_i'$ and the reflection of O with respect to line $A_{i+1}A_{i+2}$ be $O_i$ (subscripts are to be taken modulo $3$). Prove that the circumcenters of the triangles $OO_iA_i'$ ($i = 1,2,3$) are collinear.

P454 Let $\Gamma_1, \Gamma_2$ be two circles with centers $O_1, O_2$ respectively. Let $P$ be a point of intersection of $\Gamma_1$ and $\Gamma_2$. Let line $AB$ be an external common tangent to $\Gamma_1, \Gamma_2$ with $A$ on $\Gamma_1, B$ on $\Gamma_2$ and $A, B, P$ on the same side of line $O_1O_2$. There is a point $C$ on segment $O_1O_2$ such that lines $AC$ and $BP$ are perpendicular. Prove that $\angle APC=90^o$.

P459 $H$ is the orthocenter of acute $\triangle ABC$. $D,E,F$ are midpoints of sides $BC, CA, AB$ respectively. Inside $\triangle ABC$, a circle with center $H$ meets $DE$ at $P,Q, EF$ at $R,S, FD$ at $T,U$. Prove that $CP=CQ=AR=AS=BT=BU$.

P461 Inside rectangle $ABC$D, there is a circle. Points $W, X, Y, Z$ are on the circle such that lines $AW, BX, CY, DZ$ are tangent to the circle. If $AW=3, BX=4, CY=5$, then find $DZ$ with proof.

P465 Points $A, E, D, C, F, B$ lie on a circle $\Gamma$ in clockwise order. Rays $AD, BC$, the tangents to $\Gamma$ at $E$ and at $F$ pass through $P$. Chord $EF$ meets chords $AD$ and $BC$ at $M$ and $N$ respectively. Prove that lines $AB, CD, EF$ are concurrent.

P468 . Let $ABCD$ be a cyclic quadrilateral satisfying $BC>AD$ and $CD>AB$. $E, F$ are points on chords $BC, CD$ respectively and M is the midpoint of $EF$. If $BE=AD$ and $DF=AB$, then prove that $ BM \perp DM$ .

P474 Quadrilateral $ABCD$ is convex and lines $AB$, $CD$ are not parallel. Circle $\Gamma$ passes through $A, B$ and side $CD$ is tangent to $\Gamma$ at $P$. Circle $L$ passes through $C, D$ and side $AB$ is tangent to $L$ at $Q$. Circles $\Gamma$ and $L$ intersect at $E$ and $F$. Prove that line $EF$ bisects line segment PQ if and only if lines $AD, BC$ are parallel.

P477 In $\triangle ABC$, points $D, E$ are on sides $AC, AB$ respectively. Lines $BD, CE$ intersect at a point $P$ on the bisector of $\angle BAC$. Prove that quadrilateral $ADPE$ has an inscribed circle if and only if $AB=AC$.

P482 On $\triangle ABD, C$ is a point on side $BD$ with $C\ne B,D$. Let $K_1$ be the circumcircle of $\triangle ABC$. Line $AD$ is tangent to $K_1$ at $A$. A circle $K_2$ passes through $A$ and $D$ and line $BD$ is tangent to $K_2$ at $D$. Suppose $K_1$ and $K_2$ intersect at $A$ and $E$ with $E$ inside $\triangle ACD$. Prove that $EB/EC= (AB/AC)^3$ .

P487 Let $ABCD$ and $PSQR$ be squares with point $P$ on side $AB$ and $AP>PB$. Let point $Q$ be outside square $ABCD$ such that $AB \perp PQ$ and $AB=2PQ$. Let $DRME$ and $CSNF$ be squares as shown below. Prove $Q$ is the midpoint of line segment $MN$.

P490 For a parallelogram $ABCD$, it is known that $\triangle ABD$ is acute and $AD=1$. Prove that the unit circles with centers $A, B, C, D$ cover $ABCD$ if and only if $AB \le cos\angle BAD + 3sin\angle BAD$.

P492 In convex quadrilateral $ADBE$, there is a point $C$ within $\triangle ABE$ such that $\angle EAD+\angle CAB=180^o =\angle EBD+\angle CBA$. Prove that $\angle ADE=\angle BDC$.

P499 Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. Let $\Gamma$ be the escribed circle of $\triangle ABC$ meeting side $BC$ at $L$. Let line $AB$ meet $\Gamma$ at $M$ and line $AC$ meet $\Gamma$ at $N$. If the midpoint of line segment $MN$ lies on the circumcircle of $\triangle ABC$, then prove that points $O, I, L$ are collinear.

P502 Let $O$ be the center of the circumcircle of acute $\triangle ABC$. Let $P$ be a point on arc $BC$ so that $A, P$ are on opposite sides of side $BC$. Point $K$ is on chord $AP$ such that $BK$ bisects $\angle ABC$ and $\angle AKB > 90^o$. The circle $\Omega$ passing through $C, K, P$ intersect side $AC$ at $D$. Line $BD$ meets $\Omega$ at $E$ and line $PE$ meets side AB at $F$. Prove that $\angle ABC =2\angle FCB$.

P509 In $\triangle ABC$, the angle bisector of $\angle CAB$ intersects $BC$ at a point $L$. On sides $AC, AB$, there are points $M, N$ respectively such that lines $AL, BM, CN$ are concurrent and $\angle AMN=\angle ALB$. Prove that $\angle NML=90^o$.

P512 Let $AD, BE, CF$ be the altitudes of acute $\triangle ABC$. Points $P$ and $Q$ are on segments $DF$ and $EF$ respectively. If $\angle PAQ=\angle DAC$, then prove that $AP$ bisects $\angle FPQ$.

P518 Let $I$ be the incenter and $AD$ be a diameter of the circumcircle of $\triangle ABC$. Let point $E$ be on the ray $BA$ and point $F$ be on the ray $CA$. If the lengths of $BE$ and $CF$ are both equal to the semiperimeter of $\triangle ABC$, then prove that lines $EF$ and $DI$ are perpendicular.

P524 In $\triangle ABC$ with centroid $G$ , $M$ and $N$ are the midpoints of $AB$ and $AC$, and the tangents from $M$ and $N$ to the circumcircle of $\triangle AMN$ meet $BC$ at $R$ and $S$ , respectively. Point $X$ lies on $BC$ satisfying $\angle CAG=\angle BAX$ . Show that $GX$ is the radical axis of the circumcircles of $\triangle BMS$ and $\triangle CNR$.

P9 On sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ with $AB < CD$, locate points $F$ and $E$, respectively, such that $\frac{AF}{FD}=\frac{BE}{EC} =\frac{AB}{CD}$. Suppose $EF$ when extended beyond $F$ meets line $BA$ at $P$ and meets line $CD$ at $Q$. Show that $\angle BPE = \angle CQE$.

P14 Suppose $\triangle ABC, \triangle A'B'C'$ are (directly) similar to each other and $\triangle AA'A'', \triangle BB'B''. \triangle CC'C'' $ are also (directly) similar to each other. Show that \triangle $A''B''C''$ is (directly) similar to $\triangle ABC$.

P19 Suppose $A$ is a point inside a given circleand is different from the canter. Consider all chords (excludinh the diameter) passing through $A$. What is the locus of intersections of the tangent lines at the endpoints of these chords?

P27 Let $ABCD$ be a cyclic quadrilateral and let $I_A, I_B, I_C, I_D$ be the incenters of $\triangle BCD, \triangle ACD, \triangle ABD, \triangle ABC$, respectively. Show that $I_AI_BI_CI_D$ is a rectangle.

P48 Squares $ABDE$ and $BCFG$ are drawn outside of triangle $ABC$. Prove that triangle $ABC$ is isosceles if DG is parallel to $AC$.

P53 For $\triangle ABC$, define $A'$ on $BC$ so that $AB + BA' = AC + CA'$ and similarly define $B'$ on $CA$ and $C'$ on $AB$. Show that $AA', BB', CC'$ are concurrent.

P84 Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$ of $\triangle ABC$, respectively. Draw an arbitrary line through $A$. Let $Q$ and $R$ be the feet of the perpendiculars from $B$ and $C$ to this line, respectively. Find the locus of the intersection $P$ of the lines $QM$ and $RN$ as the line rotates about $A$.

P115 Find the locus of the points $P$ in the plane of an equilateral triangle $ABC$ for which the triangle formed with lengths $PA, PB$ and $PC$ has constant area.

by Mohammed Aassila, Universite Louis Pasteur, Strasbourg, France

P158 Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D$ be a point on $B$C such that $BD = 2DC$ and let $P$ be a point on $AD$ such that $\angle BAC = \angle BPD$. Prove that $\angle BAC = 2 \angle DPC$.

P164 Let $O$ be the center of the excircle of triangle $ABC$ opposite $A$. Let $M$ be the midpoint of $AC$ and let $P$ be the intersection of lines $MO$ and $BC$. Prove that if $\angle BAC = 2 \angle ACB$, then $AB = BP$.

P168 Let $AB$ and $CD$ be nonintersecting chords of a circle and let $K$ be a point on $CD$. Construct (with straightedge and compass) a point $P$ on the circle such that $K$ is the midpoint of the part of segment $CD$ lying inside triangle $ABP$.

P174 Let $M$ be a point inside acute triangle $ABC$. Let $A', B', C'$ be the mirror images of $M$ with respect to $BC, CA, AB$, respectively. Determine (with proof) all points $M$ such that $A, B, C, A', B', C'$ are concyclic.

P179 Prove that in any triangle, a line passing through the incenter cuts the perimeter of the triangle in half if and only if it cuts the area of the triangle in half .

P184 Let $ABCD$ be a rhombus with $\angle B = 60^o$ . $M$ is a point inside $\triangle ADC$ such that $\angle AMC = 120^o$ . Let lines $BA$ and $CM$ intersect at $P$ and lines $BC$ and $AM$ intersect at $Q$. Prove that $D$ lies on the line $PQ$.

P188 The line $S$ is tangent to the circumcircle of acute triangle $ABC$ at $B$. Let $K$ be the projection of the orthocenter of triangle $ABC$ onto line $S$. Let $L$ be the midpoint of side $AC$. Show that triangle $BKL$ is isosceles

P194 A circle with center $O$ is internally tangent to two circles inside it, with centers $O_1$ and $O_2$, at points $S$ and $T$ respectively. Suppose the two circles inside intersect at points $M, N$ with $N$ closer to $ST$. Show that $S, N, T$ are collinear if and only if $\frac{SO_1}{OO_1} = \frac{OO_2}{TO_2}$.

by Achilleas Pavlos Porfyriadis, American College of Thessaloniki “Anatolia”,

Thessaloniki, Greece

Thessaloniki, Greece

P198 In a triangle $ABC, AC = BC$. Given is a point $P$ on side $AB$ such that $\angle ACP = 30^o$. In addition, point $Q$ outside the triangle satisfies $\angle CPQ=\angle CPA + \angle APQ = 78^o$. Given that all angles of triangles $ABC$ and $QPB$, measured in degrees, are integers, determine the angles of these two triangles.

P202 For triangle $ABC$, let $D, E, F$ be the midpoints of sides $AB, BC, CA$, respectively. Determine which triangles $ABC$ have the property that triangles $ADF, BED, CFE$ can be folded above the plane of triangle $DEF$ to form a tetrahedron with $AD$ coincides with $BD,BE$ coincides with $CE, CF$ coincides with $AF$.

due to LUK Mee Lin, La Salle College

P208 In $\triangle ABC, AB > AC > BC$.Let $D$ be a point on the minor arc $BC$ of the circumcircle of $\triangle ABC$. Let $O$ be the circumcenter of $\triangle ABC$. Let $E, F$ be the intersection points of line $AD$ with the perpendiculars from $O$ to $AB, AC$, respectively. Let $P$ be the intersection of lines $BE$ and $CF$. If $PB = PC + PO$, then find $\angle BAC$ with proof.

P214 Let the inscribed circle of triangle $ABC$ be tangent to sides $AB, BC$ at $E$ and $F$ respectively. Let the angle bisector of $\angle CAB$ intersect segment $EF$ at $K$. Prove that $\angle CKA$ is a right angle.

P228 In $\triangle ABC, M$ is the foot of the perpendicular from $A$ to the angle bisector of $\angle BCA$. $N$ and $L$ are respectively the feet of perpendiculars from $A$ and $C$ to the bisector of $\angle ABC$. Let $F$ be the intersection of lines $MN$ and $AC$. Let $E$ be the intersection of lines $BF$ and $CL$. Let $D$ be the intersection of lines $BL$ and $AC$. Prove that lines $DE$ and $MN$ are parallel.

P245 $ABCD$ is a concave quadrilateral such that $\angle BAD =\angle ABC =\angle CDA = 45^o$. Prove that $AC = BD$.

P248 Let $ABCD$ be a convex quadrilateral such that line $CD$ is tangent to the circle with side $AB$ as diameter. Prove that line $AB$ is tangent to the circle with side $CD$ as diameter if and only if lines $BC$ and $AD$ are parallel.

P253 Suppose the bisector of $\angle BAC$ intersect the arc opposite the angle on the circumcircle of $\triangle ABC$ at $A_1$. Let $B_1$ and $C_1$ be defined similarly. Prove that the area of $\triangle A_1B_1C_1$ is at least the area of $\triangle ABC$.

P262 Let $O$ be the center of the circumcircle of $\triangle ABC$ and let $AD$ be a diameter. Let the tangent at $D$ to the circumcircle intersect line $BC$ at $P$. Let line $PO$ intersect lines $AC, AB$ at $M, N$ respectively. Prove that $OM = ON$.

P268, 337 In triangle $ABC, \angle ABC = \angle ACB = 40^o$. Points $P$ and $Q$ are inside the triangle such that $\angle PAB = \angle QAC = 20^o$ and $\angle PCB = \angle QCA = 10^o$ . Must $B, P, Q$ be collinear? Give a proof.

P272 $\triangle ABC$ is equilateral. Find the locus of all point $Q$ inside the triangle such that $ \angle QAB + \angle QBC + \angle QCA =90^o$ .

P278 Line segment $SA$ is perpendicular to the plane of the square $ABCD$. Let $E$ be the foot of the perpendicular from $A$ to line segment $SB$. Let $P, Q, R$ be the midpoints of $SD, BD, CD$ respectively. Let $M, N$ be on line segments $PQ, PR$ respectively. Prove that $AE$ is perpendicular to $MN$.

P288 (also) Let $H$ be the orthocenter of triangle $ABC$. Let $P$ be a point in the plane of the triangle such that $P$ is different from $A, B, C$. Let $L, M, N$ be the feet of the perpendiculars from $H$ to lines $PA, PB, PC$ respectively. Let $X, Y, Z$ be the intersection points of lines $LH, MH, NH$ with lines $BC, CA, AB$ respectively. Prove that $X, Y, Z$ are on a line perpendicular to line $PH$.

P293 Let $CH$ be the altitude of triangle $ABC$ with $\angle ACB = 90^o$. The bisector of $\angle BAC$ intersects $CH, CB$ at $P, M$ respectively. The bisector of $\angle ABC$ intersects $CH, CA$ at $Q, N$ respectively. Prove that the line passing through the midpoints of $PM$ and $QN$ is parallel to line $AB$.

P298 The diagonals of a convex quadrilateral $ABCD$ intersect at $O$. Let $M_1$ and $M_2$ be the centroids of $\triangle AOB$ and $\triangle COD$ respectively. Let $H_1$ and $H_2$ be the orthocenters of $\triangle BOC$ and $\triangle DOA$ respectively. Prove that $M_1M_2 \perp H_1H_2$.

305 A circle $\Gamma_2$ is internally tangent to the circumcircle $\Gamma_1$ of $\triangle PAB$ at $P$ and side $AB$ at $C$. Let $E, F$ be the intersection of $\Gamma_2$ with sides $PA, PB$ respectively. Let $EF$ intersect $PC$ at $D$. Lines $PD, AD$ intersect $\Gamma_1$ again at $G, H$ respectively. Prove that $F, G, H$ are collinear.

309 (ISL 2002 G7) In acute triangle $ABC, AB > AC$. Let $H$ be the foot of the perpendicular from $A$ to $BC$ and $M$ be the midpoint of $AH$. Let $D$ be the point where the incircle of $\triangle ABC$ is tangent to side $BC$. Let line $DM$ intersect the incircle again at $N$. Prove that $\angle BND = \angle CND$.

313 In $\triangle ABC, AB < AC$ and $O$ is its circumcenter. Let the tangent at $A$ to the circumcircle cut line $BC$ at $D$. Let the perpendicular lines to line $BC$ at $B$ and $C$ cut the perpendicular bisectors of sides $AB$ and $AC$ at $E$ and $F$ respectively. Prove that $D, E, F$ are collinear.

321 Let $AA', BB'$ and $CC'$ be three non-coplanar chords of a sphere and let them all pass through a common point P inside the sphere. There is a (unique) sphere $S_1$ passing through $A, B, C, P$ and a (unique) sphere $S_2$ passing through $A', B', C', P$. If $S_1$ and $S_2$ are externally tangent at $P$, then prove that $AA'=BB'=CC'$.

324 $ADPE$ is a convex quadrilateral such that $\angle ADP = \angle AEP$. Extend side $AD$ beyond $D$ to a point $B$ and extend side $AE$ beyond $E$ to a point $C$ so that $\angle DPB = \angle EPC$. Let $O_1$ be the circumcenter of $\triangle ADE$ and let $O_2$ be the circumcenter of $\triangle ABC$. If the circumcircles of $\triangle ADE$ and $\triangle ABC$ are not tangent to each other, then prove that line $O_1O_2$ bisects line segment $AP$.

332 Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. Let $BD$ bisect $OC$ perpendicularly. On diagonal $AC$, choose the point $P$ such that $PC=OC$. Let line $BP$ intersect line $AD$ and the circumcircle of $ABCD$ at $E$ and $F$ respectively. Prove that $PF$ is the geometric mean of $EF$ and $BF$ in length.

344 (Bulgaria 97) $ABCD$ is a cyclic quadrilateral. Let $M, N$ be midpoints of diagonals $AC, BD$ respectively. Lines $BA, CD$ intersect at $E$ and lines $AD, BC$ intersect at $F$. Prove that $ \big| \frac{BD}{AC}-\frac{AC}{BD}\big|= \frac{2MN}{EF}$

__geometry__

__problems from problem corner__

without aops links (no contest's problems)

P358 $ABCD$ is a cyclic quadrilateral with $AC$ intersects $BD$ at $P$. Let $E, F, G, H$ be the feet of perpendiculars from $P$ to sides $AB, BC, CD, DA$ respectively. Prove that lines $EH, BD, FG$ are concurrent or are parallel.

P363 Extend side $CB$ of triangle $ABC$ beyond $B$ to a point $D$ such that $DB=AB$. Let $M$ be the midpoint of side $AC$. Let the bisector of $\angle ABC$ intersect line $DM$ at $P$. Prove that $\angle BAP =\angle ACB$.

P369 $ABC$ is a triangle with $BC > CA > AB$. $D$ is a point on side $BC$ and $E$ is a point on ray $BA$ beyond $A$ so that $BD=BE=CA$. Let $P$ be a point on side $AC$ such that $E, B, D, P$ are concyclic. Let $Q$ be the intersection point of ray $BP$ and the circumcircle of $\triangle ABC$ different from $B$. Prove that $AQ+CQ=BP$ .

P374 $O$ is the circumcenter of acute $\triangle ABC$ and $T$ is the circumcenter of $\triangle AOC$. Let $M$ be the midpoint of side $AC$. On sides $AB$ and $BC$, there are points $D$ and $E$ respectively such that $\angle BDM=\angle BEM=\angle ABC$. Prove that $BT\perp DE$.

P379 Let $\ell$ be a line on the plane of $\triangle ABC$ such that $\ell$ does not intersect the triangle and none of the lines $AB, BC, CA$ is perpendicular to $\ell$. Let $A', B', C'$ be the feet of the perpendiculars from $A, B, C$ to $\ell$ respectively. Let $A'', B'', C''$ be the feet of the perpendiculars from $A', B',C'$ to lines $BC, CA, AB$ respectively. Prove that lines $A'A'', B'B'', C'C''$ are concurrent.

P383 Let $O$ and $I$ be the circumcenter and incenter of $\triangle ABC$ respectively. If $AB\ne AC$, points $D, E$ are midpoints of $AB, AC$ respectively and $BC=(AB+AC)/2$, then prove that the line $OI$ and the bisector of $\angle CAB$ are perpendicular .

P388 In $\triangle ABC, \angle BAC=30^o$ and $\angle ABC=70^o$. There is a point $M$ lying inside $\triangle ABC$ such that $\angle MAB= \angle MCA=20^o$. Determine $\angle MBA$ (with proof).

P394 Let $O$ and $H$ be the circumcenter and orthocenter of acute $\triangle ABC$. The bisector of $\angle BAC$ meets the circumcircle $\Gamma$ of $\triangle ABC$ at $D$. Let $E$ be the mirror image of $D$ with respect to line $BC$. Let $F$ be on $\Gamma$ such that $DF$ is a diameter. Let lines $AE$ and $FH$ meet at $G$. Let $M$ be the midpoint of side $BC$. Prove that $GM\perp A$F.

P399 Let $ABC$ be a triangle for which $\angle BAC=60^o$. Let P be the point of intersection of the bisector of $\angle ABC$ and the side $AC$. Let $Q$ be the point of intersection of the bisector of $\angle ACB$ and the side $AB$. Let $r_1$ and $r_2$ be the radii of the incircles of triangles $ABC$ and $APQ$ respectively. Determine the radius of the circumcircle of triangle $APQ$ in terms of $r_1$ and $r_2$ with proof.

P404 Let $I$ be the incenter of acute $\triangle ABC$. Let $\Gamma$ be a circle with center $I$ that lies inside $\triangle ABC$. $D, E, F$ are the intersection points of circle $\Gamma$ with the perpendicular rays from $I$ to sides $BC, CA, AB$ respectively. Prove that lines $AD, BE, CF$ are concurrent.

P407 Three circles $S, S_1, S_2$ are given in a plane. $S_1$ and $S_2$ touch each other externally, and both of them touch $S$ internally at $A_1$ and at $A_2$ respectively. Let $P$ be one of the two points where the common internal tangent to $S_1$ and $S_2$ meets $S$. Let $B_i$ be the intersection points of $PA_i$ and $S_i$ ($i=1,2$). Prove that line $B_1B_2$ is a common tangent to $S_1$ and $S_2$.

P412 $\triangle ABC$ is equilateral and points $D, E, F$ are on sides $BC, CA, AB$ respectively. If $\angle BAD +\angle CBE + \angle ACF = 120^o$,

then prove that $\triangle BAD, \triangle CBE$ and $\triangle ACF$ cover $\triangle ABC$.

P415 [also]Given a triangle $ABC$ such that $\angle BAC=103^o$ and $\angle ABC=51^o$. Let $M$ be a point inside ΔABC such that $\angle MAC=30^o$ and $\angle MCA=13^o$. Find $\angle MBC$ with proof, without trigonometry.

Due to Apostolos Manoloudis, Piraeus, Greece

P418 Point $M$ is the midpoint of side $AB$ of acute $\triangle ABC$. Points $P$ and $Q$ are the feet of perpendicular from $A$ to side $BC$ and from $B$ to side $AC$ respectively. Line $AC$ is tangent to the circumcircle of $\triangle BMP$. Prove that line $BC$ is tangent to the circumcircle of $\triangle AMQ$.

P421 For every acute triangle $ABC$, prove that there exists a point $P$ inside the circumcircle $\omega$ of ∆ABC such that if rays $AP, BP, CP$ intersect $\omega$ at $D, E, F$, then $DE: EF: FD = 4:5:6$.

P428 Let $A_1A_2A_3A_4$ be a convex quadrilateral. Prove that the nine point circles of $\triangle A_1A_2A_3, \triangle A_2A_3A_4, \triangle A_3A_4A_1$ and $\triangle A_4A_1A_2$ pass through a common point.

P434 Let $O$ and $H$ be the circumcenter and orthocenter of $\triangle ABC$ respectively. Let $D$ be the foot of perpendicular from $C$ to side $AB$. Let $E$ be a point on line $BC$ such that $ED \perp OD$. If the circumcircle of $\triangle BCH$ intersects side $AB$ at $F$, then prove that points $E, F, H$ are collinear.

P439 In acute triangle $ABC, T$ is a point on the altitude $AD$ (with $D$ on side $BC$). Lines $BT$ and $AC$ intersect at $E$, lines $CT$ and $AB$ intersect at $F$, lines $EF$ and $AD$ intersect at $G$. A line $\ell$ passing through $G$ intersects side $AB$, side $AC$, line $BT$, line $CT$ at $M, N, P, Q$ respectively. Prove that $\angle MDQ =\angle NDP$.

P444 Let $D$ be on side $BC$ of equilateral triangle $ABC$. Let $P$ and $Q$ be the incenters of $\triangle ABD$ and $\triangle ACD$ respectively. Let $E$ be the point so that $\triangle EPQ$ is equilateral and $D, E$ are on opposite sides of line $PQ$. Prove that lines $BC$ and $DE$ are perpendicular.

P450 Let $A_1A_2A_3$ be a triangle with no right angle and $O$ be its circumcenter. For $i = 1,2,3$, let the reflection of $A_i$ with respect to $O$ be $A_i'$ and the reflection of O with respect to line $A_{i+1}A_{i+2}$ be $O_i$ (subscripts are to be taken modulo $3$). Prove that the circumcenters of the triangles $OO_iA_i'$ ($i = 1,2,3$) are collinear.

proposed by Michel Bataille

P454 Let $\Gamma_1, \Gamma_2$ be two circles with centers $O_1, O_2$ respectively. Let $P$ be a point of intersection of $\Gamma_1$ and $\Gamma_2$. Let line $AB$ be an external common tangent to $\Gamma_1, \Gamma_2$ with $A$ on $\Gamma_1, B$ on $\Gamma_2$ and $A, B, P$ on the same side of line $O_1O_2$. There is a point $C$ on segment $O_1O_2$ such that lines $AC$ and $BP$ are perpendicular. Prove that $\angle APC=90^o$.

P459 $H$ is the orthocenter of acute $\triangle ABC$. $D,E,F$ are midpoints of sides $BC, CA, AB$ respectively. Inside $\triangle ABC$, a circle with center $H$ meets $DE$ at $P,Q, EF$ at $R,S, FD$ at $T,U$. Prove that $CP=CQ=AR=AS=BT=BU$.

P461 Inside rectangle $ABC$D, there is a circle. Points $W, X, Y, Z$ are on the circle such that lines $AW, BX, CY, DZ$ are tangent to the circle. If $AW=3, BX=4, CY=5$, then find $DZ$ with proof.

P465 Points $A, E, D, C, F, B$ lie on a circle $\Gamma$ in clockwise order. Rays $AD, BC$, the tangents to $\Gamma$ at $E$ and at $F$ pass through $P$. Chord $EF$ meets chords $AD$ and $BC$ at $M$ and $N$ respectively. Prove that lines $AB, CD, EF$ are concurrent.

P468 . Let $ABCD$ be a cyclic quadrilateral satisfying $BC>AD$ and $CD>AB$. $E, F$ are points on chords $BC, CD$ respectively and M is the midpoint of $EF$. If $BE=AD$ and $DF=AB$, then prove that $ BM \perp DM$ .

P474 Quadrilateral $ABCD$ is convex and lines $AB$, $CD$ are not parallel. Circle $\Gamma$ passes through $A, B$ and side $CD$ is tangent to $\Gamma$ at $P$. Circle $L$ passes through $C, D$ and side $AB$ is tangent to $L$ at $Q$. Circles $\Gamma$ and $L$ intersect at $E$ and $F$. Prove that line $EF$ bisects line segment PQ if and only if lines $AD, BC$ are parallel.

P477 In $\triangle ABC$, points $D, E$ are on sides $AC, AB$ respectively. Lines $BD, CE$ intersect at a point $P$ on the bisector of $\angle BAC$. Prove that quadrilateral $ADPE$ has an inscribed circle if and only if $AB=AC$.

P482 On $\triangle ABD, C$ is a point on side $BD$ with $C\ne B,D$. Let $K_1$ be the circumcircle of $\triangle ABC$. Line $AD$ is tangent to $K_1$ at $A$. A circle $K_2$ passes through $A$ and $D$ and line $BD$ is tangent to $K_2$ at $D$. Suppose $K_1$ and $K_2$ intersect at $A$ and $E$ with $E$ inside $\triangle ACD$. Prove that $EB/EC= (AB/AC)^3$ .

P487 Let $ABCD$ and $PSQR$ be squares with point $P$ on side $AB$ and $AP>PB$. Let point $Q$ be outside square $ABCD$ such that $AB \perp PQ$ and $AB=2PQ$. Let $DRME$ and $CSNF$ be squares as shown below. Prove $Q$ is the midpoint of line segment $MN$.

P490 For a parallelogram $ABCD$, it is known that $\triangle ABD$ is acute and $AD=1$. Prove that the unit circles with centers $A, B, C, D$ cover $ABCD$ if and only if $AB \le cos\angle BAD + 3sin\angle BAD$.

P492 In convex quadrilateral $ADBE$, there is a point $C$ within $\triangle ABE$ such that $\angle EAD+\angle CAB=180^o =\angle EBD+\angle CBA$. Prove that $\angle ADE=\angle BDC$.

P499 Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. Let $\Gamma$ be the escribed circle of $\triangle ABC$ meeting side $BC$ at $L$. Let line $AB$ meet $\Gamma$ at $M$ and line $AC$ meet $\Gamma$ at $N$. If the midpoint of line segment $MN$ lies on the circumcircle of $\triangle ABC$, then prove that points $O, I, L$ are collinear.

P502 Let $O$ be the center of the circumcircle of acute $\triangle ABC$. Let $P$ be a point on arc $BC$ so that $A, P$ are on opposite sides of side $BC$. Point $K$ is on chord $AP$ such that $BK$ bisects $\angle ABC$ and $\angle AKB > 90^o$. The circle $\Omega$ passing through $C, K, P$ intersect side $AC$ at $D$. Line $BD$ meets $\Omega$ at $E$ and line $PE$ meets side AB at $F$. Prove that $\angle ABC =2\angle FCB$.

P509 In $\triangle ABC$, the angle bisector of $\angle CAB$ intersects $BC$ at a point $L$. On sides $AC, AB$, there are points $M, N$ respectively such that lines $AL, BM, CN$ are concurrent and $\angle AMN=\angle ALB$. Prove that $\angle NML=90^o$.

P512 Let $AD, BE, CF$ be the altitudes of acute $\triangle ABC$. Points $P$ and $Q$ are on segments $DF$ and $EF$ respectively. If $\angle PAQ=\angle DAC$, then prove that $AP$ bisects $\angle FPQ$.

P518 Let $I$ be the incenter and $AD$ be a diameter of the circumcircle of $\triangle ABC$. Let point $E$ be on the ray $BA$ and point $F$ be on the ray $CA$. If the lengths of $BE$ and $CF$ are both equal to the semiperimeter of $\triangle ABC$, then prove that lines $EF$ and $DI$ are perpendicular.

P524 In $\triangle ABC$ with centroid $G$ , $M$ and $N$ are the midpoints of $AB$ and $AC$, and the tangents from $M$ and $N$ to the circumcircle of $\triangle AMN$ meet $BC$ at $R$ and $S$ , respectively. Point $X$ lies on $BC$ satisfying $\angle CAG=\angle BAX$ . Show that $GX$ is the radical axis of the circumcircles of $\triangle BMS$ and $\triangle CNR$.

(Anderw WU)

P528 Let points $O$ and $H$ be the circumcenter and orthocenter of acute $\triangle ABC$. Let $D$ be the midpoint of side $BC$. Let $E$ be the point on the angle bisector of $\angle BAC$ such that $AE\perp HE$. Let $F$ be the point such that $AEHF$ is a rectangle. Prove that points $D, E, F$ are collinear.

__Index of Geometry Articles__

__(__with pdf links

__)__

__by Kin - Yin Li__

- A Geometry Theorem Vol.16 No. 4 (Subtended Angle Theorem)
- Angle Bisectors Bisect Arcs Vol.11 No. 2
- Casey’s Theorem Vol.16 No. 5
- Cavalieri’s Principle Vol.5 No. 1
- Concyclic Problems Vol.6 No. 1
- Coordinate Geometry Vol.5 No. 3
- Famous Geometry Theorems Vol.10 No. 3 (Brianchon, Desargues, Menelaus, Newton, Pascal)
- Geometric Transformations I Vol.13 No.2 (translation, rotation, reflection, spiral similarity)
- Geometric Transformations II Vol.13 No.3 (translation, homothety, rotation, reflection)
- Geometry via Complex Numbers Vol.9 No.1
- Homothety Vol.9 No.4
- Inversion Vol.9 No.2
- Similar Triangles via Complex Numbers Vol.1 No.3
- Perpendicular Lines Vol.12 No. 3
- Pole and Polar Vol.11 No. 4
- Polygonal Problems Vol.19 No. 4
- Power of Points Respect to Circles Vol.4 No.3
- Ptolemy's Theorem Vol.2 No.4
- Vector Geometry Vol.6 No. 5

__by K. K. Kwok__

- From How to Solve It to Problem Solving in Geometry I Vol.12 No.1
- From How to Solve It to Problem Solving in Geometry II Vol.12 No.2

__by Nguyen Ngoc Giang__

- Ptolemy’s Inequality Vol.18 No.1

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