### M. Excalibur

Here are collected 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.

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

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.

(also) (also)  $\triangle ABC$ 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 $M_1M_2 \perp H_1H_2$.

P305 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.

P309  (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 $\angle BND = \angle CND$.

P313 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.

P321 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 $AA'=BB'=CC'$.

P324 $ADPE$ 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 $AP$.

P332 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.

P344 (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 $\big| \frac{BD}{AC}-\frac{AC}{BD}\big|= \frac{2MN}{EF}$

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.

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 $\triangle AMQ$.

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

P428  (also)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 exscribed 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.

P531 $BCED$ is a convex quadrilateral such that $\angle BDC =\angle CEB= 90^o$ and $BE$ intersects $CD$ at $A$. Let $F,G$ be the midpoints of sides $DE, BC$ respectively. Let $O$ be the circumcenter of $\vartriangle BAC$. Prove that lines $AO$ and $FG$ are parallel.

P537 Distinct points $A, B, C$ are on the unit circle $\Gamma$ with center $O$ inside $\vartriangle ABC$. Suppose the feet of the perpendiculars from $O$ to sides $BC, CA,AB$ are $D, E, F$. Determine the largest value of $OD+OE+OF$.

Index of Geometry Articles