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Mathematical Excalibur - geometry
3
geometry problems from Mathematical Excalibur
4
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.
parmenides51
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9
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 $\angle BPE = \angle CQE$.
parmenides51
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14
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 $\triangle ABC$.
parmenides51
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19
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?
parmenides51
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27
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.
parmenides51
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48
Squares $ABDE$ and $BCFG$ are drawn outside of triangle $ABC$. Prove that triangle $ABC$ is isosceles if DG is parallel to $AC$.
parmenides51
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53
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.
parmenides51
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84
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 $A$.
parmenides51
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115
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.

Proposed by Mohammed Aassila, Universite Louis Pasteur, Strasbourg, France
parmenides51
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158
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 $\angle BAC = 2 \angle DPC$.
parmenides51
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164
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 $AB = BP$.
parmenides51
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168
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 $ABP$.
parmenides51
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174
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.
parmenides51
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179
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 .
parmenides51
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184
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 $PQ$.
parmenides51
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188
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
parmenides51
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194
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
parmenides51
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198
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.
parmenides51
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202
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
parmenides51
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208
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.
parmenides51
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214
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.
parmenides51
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228
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.
parmenides51
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245
$ABCD$ is a concave quadrilateral such that $\angle BAD =\angle ABC =\angle CDA = 45^o$. Prove that $AC = BD$.
parmenides51
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248
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.
parmenides51
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253
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 $\triangle ABC$.
parmenides51
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262
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 $OM = ON$.
parmenides51
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268
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.
parmenides51
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272
Given an equilateral triangle $ABC$. Find the locus of points $M$ inside the triangle such that $\angle MAB + \angle MBC + \angle MCA =\frac{\pi}{2}$
GeoMetrix
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278
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 $MN$.
parmenides51
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288
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 $PH$.
parmenides51
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293
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 $AB$.
parmenides51
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298
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 $M_1M_2 \perp H_1H_2$
parmenides51
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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.
parmenides51
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309
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 $\angle BND = \angle CND$.
parmenides51
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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.
parmenides51
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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 $AA'=BB'=CC'$.
parmenides51
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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 $AP$.
parmenides51
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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.
parmenides51
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344
$ABCD$ 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}$
parmenides51
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355
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?
parmenides51
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358
$ABCD$ 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.
parmenides51
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363
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 $\angle BAP =\angle ACB$.
parmenides51
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369
$ABC$ 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 $AQ+CQ=BP$
parmenides51
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374
$O$ 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 $BT\perp DE$.
parmenides51
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379
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.
parmenides51
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383
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 .
parmenides51
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388
In $\triangle ABC, \angle BAC=30^\circ$ and $\angle ABC=70^\circ$. There is a point $M$ lying inside $\triangle ABC$ such that $\angle MAB=\angle MCA=20^\circ.$ Determine $\angle MBA$?
sodoo
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394
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.
parmenides51
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399
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.
parmenides51
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404
Let $ I $ be the incenter of an acute $\triangle$ $ABC $.Let $ \omega $ be a circle with center $ I $ that lies inside $\triangle$ $ABC$ .$ D $, $ E $, $ F $ are the intersection points of circle $\omega $ with the perpendicular rays from $ I $ to sides $ BC $, $ CA $, $ AB $ respectively. Prove that the lines $ AD $, $ BE $, $ CF $ are concurrent.
abmax777
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407
Two circles $\omega_1$ and $\omega_2$ touch each other externally at $T$. They also touch a circle $\omega$ internally at $A_1$ and $A_2$, respectively. Let $P$ be one point of intersection of $\omega$ with the common tangent to $\omega_1$ and $\omega_2$ at $T$. The line $PA_1$ meets $\omega_1$ again at $B_1$ and the line $PA_2$ meets $\omega_2$ again at $B_2$. Prove that $B_1B_2$ is a common tangent to $\omega_1$ and $\omega_2$.
mufree
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412
$\triangle ABC$ 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 $\triangle ABC$.
parmenides51
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415
Given a triangle ABC such that ∠BAC = 103° and ∠ABC = 51°. Let M be a point inside ΔABC such that ∠MAC = 30°
and ∠MCA = 13°. Find ∠MBC with proof without trigonometry.
MariusStanean
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418
Let P and Q be the feet of altitudes from A and B of acute-angled triangle ABC, and let M be the midpoint of BA.Show that if the circumcircle of triangleBPM is tangent to the line AC then the circumcircle of triangle QMA is tangent to the line BC.
Math-lover123
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421
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 $DE: EF: FD = 4:5:6$.
parmenides51
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428
In a convex quadrilateral $ABCD,$ prove that the nine point circles of the triangles $\triangle ABC,\ \triangle BCD,\ \triangle CDA,\ \triangle DAB,$ are concurrent at one point.

Kostas Vittas.

PS. This result has been mentioned in the topic What angle?, by Jean-Louis.
vittasko
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434
Let $O$ be the circumcenter of triangle $ABC$ and $H$ be it's orthocenter where is $D$ the foot from $C$. Perpendicular line to $OD$ in the point $D$ intersects side $BC$ at the point $E$. Circumcircle of $BCH$ intersects the line $AB$ at points $B$ and $F$. Prove, that $H$, $E$, $F$ are collinear.
JANMATH111
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439
In acute triangle $\triangle{ABC}$, $T$ 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 $l$ passing through $G$ intersects side $AB$,side $AC$, line $BT$, line $CT$ at $M,N,P,Q$ respectively. Prove that $\widehat{MDQ}=\widehat{NDP}$
Aiscrim
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444
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.
MariusStanean
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450
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.

(Michel Bataille)
parmenides51
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454
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 $\angle APC=90^o$.
parmenides51
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459
$H$ 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 $CP=CQ=AR=AS=BT=BU$.
parmenides51
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461
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.
parmenides51
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465
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.
KudouShinichi
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468
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$ .
parmenides51
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474
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.
parmenides51
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477
In $\triangle ABC$, points $D, E$ are on sides $AC, AB$ respectively. Lines $BD, CE$ intersect at a point $P$ on the angle bisector of $\angle BAC$. Prove that quadrilateral $ADPE$ has an inscribed circle if and only if $AB=AC$.
parmenides51
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482
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 $EB/EC= (AB/AC)^3$ .
parmenides51
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487
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 $MN$.
https://2.bp.blogspot.com/-POcw0REvum0/W3Fwl7nLt6I/AAAAAAAAI-A/tL8LLwFrKDswuuHts9jiVx53L54yHOxRgCK4BGAYYCw/s1600/excalibur%2Bp487.png
parmenides51
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490
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 $AB \le cos\angle BAD + 3sin\angle BAD$.
parmenides51
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492
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 $\angle ADE=\angle BDC$
parmenides51
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499
Denote $I$ and $O$ as incenter and circumcenter of $\triangle ABC$. The excentre $\omega_A$ of $\triangle ABC$ tangent to $BC$ at $N$ and tangent to the extension of $AB$ and $AC$ each at $K$ and $M$. Suppose that the midpoint of $KM$ lies on the circumcircle of $\triangle ABC$. Prove that $O,I,N$ is collinear.
GorgonMathDota
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502
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 $\angle ABC =2\angle FCB$.
parmenides51
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509
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 $\angle NML=90^o$.
parmenides51
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512
Let $AA_1,BB_1,CC_1$ be the altitudes of an acute triangle $ABC$. Points $M$ and $N$ are on segments $B_1C_1$ and $C_1A_1$ respectively such that $\angle MAN=\angle B_1AA_1$. Prove that line $NA$ bisects $\angle MNC_1$.

I'm looking for a solution without trigonometry.
jgnr
view topic
518
Problem 518.
Let $I$ be the incenter and $AD$ be a diameter of the circumcircle of $ABC$. 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 $ABC$, then prove that lines $EF$ and $DI$ are perpendicular.
WolfusA
view topic
524
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 $\triangle CNR$.

Andrew WU
XbenX
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528
Let the points $O$ and $H$ be the circumcenter and orthocenter of an acute angled triangle $ABC.$ Let $D$ be the midpoint of $BC.$ 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 $D,E,F$ are collinear.
ayan.nmath
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531
$BCED$ 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.
parmenides51
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537
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 $OD+OE+OF$.
parmenides51
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a