### MR

Mathematical Reflections

Here are gonna be collected all the Euclidean Geometry problems (with or without aops links), geometry articles and the problems with solutions from the online magazine ''Mathematical Reflections''.

(2006 issues)

collected inside aops here:

Junior level
[under construction]

J6 Let $ABCD$ be a convex quadrilateral such that the sides $BC$ and $CD$ have equal lengths and $2\angle A+\angle C = 180^o$. Let $M$ be the midpoint of the line segment $BD$. Prove that $\angle MAD = \angle BAC$
by Dinu Serbanescu, ”Sf. Sava” National College, Romania

J11 Consider an arbitrary parallelogram $ABCD$ with center $O$ and let $P$ be a point different from $O$, that satisfies $PA \cdot PC = OA \cdot OC$ and $PB \cdot PD = OB\cdot OD$. Show that the sum of lengths of two of the segments $PA, PB, PC, PD$ equals the sum of lengths of the other two.
by Iurie Boreico, student, Chisinau, Moldova

J16 Consider a scalene triangle $ABC$ and let $X \in (AB)$ and $Y \in (AC)$ be two variable points such that $(BX) = (CY)$. Prove that the circumcircle of triangle $AXY$ passes through a fixed point (different from $A$).
by Liubomir Chiriac, student, Chișinău, Moldova

J23 Let $ABCDEF$ be a hexagon with parallel opposite sides, and let $FC\cap AB = X_1, FC\cap ED = X_2, AD\cap EF = Y_1, AD\cap BC = Y_2, BE\cap CD = Z_1, BE\cap AF = Z_2$. Prove that if $X_1, Y_1,Z_1$ are collinear then $X_2, Y_2,Z_2$ are also collinear and in this case the lines $X_1Y_1Z_1$ and $X_2Y_2Z_2$ are parallel.
by Santiago Cuellar

J27 Consider points $M,N$ inside the triangle $ABC$ such that $\angle BAM = \angle CAN,\angle MCA = \angle NCB,$ $\angle MBC = \angle CBN$. $M$ and $N$ are isogonal points. Suppose $BMNC$ is a cyclic quadrilateral. Denote $T$ the circumcenter of $BMNC$, prove that $MN \perp AT$.
by Ivan Borsenco, University of Texas at Dallas

J34 Let $ABC$ be a triangle and let $I$ be its incenter. Prove that at least one of $IA, IB, IC$ is greater than or equal to the diameter of the incircle of $ABC$.

by Magkos Athanasios, Kozani, Greece

Senior level
[under construction]

Circles with radii $r_1, r_2, r_3$ are externally tangent to each other. Two other circles, with radii $R$ and $r$, are tangent to all previous circles. Prove that: $Rr \ge \frac{r_1r_2r_3}{r_1 + r_2 + r_3}$
by Ivan Borsenco, University of Texas at Dallas

S8 Let $O, I$, and $r$ be the circumcenter, incenter, and inradius of a triangle $ABC$. Let $M$ be a point inside the triangle; and let $d_1,d_2, d_3,$ be the distances from $M$ to the sides $BC,AC,AB$. Prove that if $d_1\cdot d_2 \cdot d_3 \ge r^3$, then $M$ lies inside the circle with center $O$ and radius $OI$.
by Ivan Borsenco, student, Chisinau, Moldova

S15 Consider a scalene triangle $ABC$ and let $X \in AB$ and $Y \in AC$ be two variable points such that $BX = CY$ . If $\{Z\} = BY \cap CX$ and the circumcircles of $\triangle AY B$ and $\triangle AXC$ meet each other at $A$ and $K$, prove that the reflection of $K$ across the midpoint of $AZ$ belongs to a fixed line.
by Liubomir Chiriac, student, Chișinău, Moldova

S16 Let $M_1$ be a point inside triangle $ABC$ and let $M_2$ be its isogonal conjugate. Let $R$ and $r$ denote the circumradius and the inradius of the triangle. Prove that $4R^2r^2 \ge (R^2 - OM_1^2) (R^2 - OM_2^2)$

by Ivan Borsenco, student, Chișinău, Moldova

S19 Let $ABC$ be a scalene triangle. A point $P$ is called nice if $AD,BE,CF$ are concurrent, where $D,E, F$ are the projections of $P$ onto $BC, CA, AB$, respectively. Find the number of nice points that lie on the line $OI$.

by Iurie Boreico, Moldova and Ivan Borsenco, University of Texas at Dallas

S24 Let $ABC$ be an acute-angled triangle inscribed in a circle $C$. Consider all equilateral triangles $DEF$ with vertices on $C$. The Simpson lines of $D,E, F$ with respect to the triangle $ABC$ form a triangle $T$ . Find the greatest possible area of this triangle.

by Iurie Boreico, Moldova and Ivan Borsenco, University of Texas at Dallas

S26 Consider a triangle $ABC$ and let $I_a$ be the center of the circle that touches the side $BC$ at $A'$ and the extensions of sides $AB$ and $AC$ at $C'$  and $B'$ , respectively. Denote by $X$ the second intersections of the line $A' B'$  with the circle with center $B$ and radius $BA'$  and by $K$ the midpoint of $CX$. Prove that $K$ lies on the midline of the triangle $ABC$ corresponding to $AC$.
by Liubomir Chiriac, Princeton University

S28  Let $M$ be a point in the plane of triangle $ABC$. Find the minimum of $MA^3 +MB^3 +MC^3- \frac{3}{ 2}R \cdot MH^2$, where $H$ is the orthocenter and $R$ is the circumradius of the triangle $ABC$.
by Hung Quang Tran, Hanoi, Vietnam

S31 Let $ABC$ be a triangle and let $P, Q,R$ be three points lying inside $ABC$. Suppose quadrilaterals $ABPQ, ACPR, BCQR$ are concyclic. Prove that if the radical center of these circles is the incenter $I$ of triangle $ABC$, then the Euler line of the triangle $PQR$ coincides with $OI$, where $O$ is the circumcenter of triangle $ABC$.

by Ivan Borsenco, University of Texas at Dallas

S34 Let $ABC$ be an equilateral triangle and let $P$ be a point on its circumcircle. Find all positive integers n such that $PA^n + PB^n + PC^n$ does not depend upon $P$.

by Oleg Mushkarov, Bulgarian Academy of Sciences, Sofia

S36 Let $P$ be a point in the plane of a triangle $ABC$, not lying on the lines $AB,BC$, or $CA$. Denote by $A_b,A_c$ the intersections of the parallels through $A$ to the lines $PB, PC$ with the line $BC$. Define analogously $B_a,B_c,C_a,C_b$. Prove that $A_b,A_c,B_a,B_c,C_a,C_b$ lie on the same conic.

[under construction]

O1 A circle centered at $O$ is tangent to all sides of the convex quadrilateral $ABCD$. The rays $BA$ and $CD$ intersect at $K$, the rays $AD$ and $BC$ intersect at $L$. The points $X, Y$ are considered on the line segments $OA,OC$, respectively. Prove that $\angle XKY = \frac{1}{2} \angle AKC$ if and only if $\angle XLY = \frac{1}{2} \angle ALC$.

by Pavlo Pylyavskyy, MIT

O4  Let $AB$ be a diameter of the circle $\Gamma$ and let $C$ be a point on the circle, different from $A$ and $B$. Denote by $D$ the projection of $C$ on $AB$ and let $\omega$ be a circle tangent to $AD, CD$, and $\Gamma$, touching $\Gamma$ at $X$. Prove that the angle bisectors of $\angle AXB$ and $\angle ACD$ meet on $AB$.
by Liubomir Chiriac, Princeton

O7 In the convex hexagon $ABCDEF$ the following equalities hold:
$AD = BC + EF, BE = AF + CD, CF = AB + DE$ .
Prove that $\frac{AB}{DE}=\frac{CD}{AF}=\frac{EF}{BC}$.
by Nairi Sedrakyan, Armenia

O13 Let $ABC$ be a triangle and $P$ be an arbitrary point inside the triangle. Let $A',B',C'$, respectively, be the intersections of $AP, BP$, and $CP$ with the triangle’s sides. Through $P$ we draw a line perpendicular to $PA$ that intersects $BC$ at $A_1$. We define $B_1$ and $C_1$ analogously. Let $P'$ be the isogonal conjugate of the point $P$ with respect to triangle $A'B'C'$. Show that $A_1,B_1$, and $C_1$ lie on a line $\ell$ that is perpendicular to $PP'$.

by Khoa Lu Nguyen, Sam Houston High School, Houston, Texas.

O16 Let $ABC$ be an acute-angled triangle. Let $\omega$ be the center of the nine point circle and let $G$ be its centroid. Let $A',B',C',A'',B'',C''$ be the projections of $\omega$  and $G$ on the corresponding sides. Prove that the perimeter of $A''B''C''$ is not less than the perimeter of $A'B'C'$.

by Iurie Boreico, student, Chișinău, Moldova

O20 The incircle of triangle $ABC$ touches $AC$ at $E$ and $BC$ at $D$. The excircle corresponding to $A$ touches the side $BC$ at $A_1$ and the extensions of $AB, AC$ at $C_1$ and $B_1$, respectively. Let $DE \cap A_1B_1 = L$. Prove that $L$ lies on the circumcircle of triangle $A_1BC_1$.
Liubomir Chiriac, Princeton University

O22 Consider a triangle $ABC$ and points $P,Q$ in its plane. Let $A_1,B_1,C_1$ and $A_2,B_2,C_2$ be cevians in this triangle. Denote by $U, V,W$ the second intersections of circles $(AA_1A_2), (BB_1B_2), (CC_1C_2)$ with circle $(ABC)$, respectively. Let $X$ be the point of intersection of $AU$ with $BC$. Similarly define $Y$ and $Z$. Prove that $X, Y,Z$ are collinear.

Khoa Lu Nguyen, M.I.T and Ivan Borsenco, University of Texas at Dallas

O23 Let $ABC$ be a triangle and let $A_1,B_1,C_1$ be the points where the angle bisectors of $A,B$ and $C$ meet the circumcircle of triangle $ABC$, respectively. Let $M_a$ be the midpoint of the segment connecting the intersections of segments $A_1B_1$ and $A_1C_1$ with segment $BC$. Define $M_b$ and $M_c$ analogously. Prove that $AM_a,BM_b$, and $CM_c$ are concurrent if and only if $ABC$ is isosceles.
by Dr. Zuming Feng, Phillips Exeter Academy, New Hampshire

O26 Consider a triangle $ABC$ and let $O$ be its circumcenter. Denote by $D$ the foot of the altitude from $A$ and by $E$ the intersection of $AO$ and $BC$.  Suppose tangents to the circumcircle of triangle $ABC$ at $B$ and $C$ intersect at $T$ and that $AT$ intersects this circumcircle at $F$. Prove that the circumcircles of triangles $DEF$ and $ABC$ are tangent.

by Ivan Borsenco, University of Texas at Dallas

O33 Let $ABC$ be a triangle with cicrumcenter $O$ and incenter $I$.  Consider a point $M$ lying on the small arc $BC$. Prove that $AM + 2OI \ge MB +MC \ge MA - 2OI$

by Hung Quang Tran, Ha Noi University, Vietnam

[2006-2019 under construction]

U47 Let $P$ be arbitrary point inside equilateral triangle $ABC$. Find the minimum value of
$\frac{1}{PA} + \frac{1}{PB} + \frac{1}{PC}$

by Hung Quang Tran, Ha Noi National University, Vietnam

U68 In the plane consider two lines $d_1$ and $d_2$ and let $B,C \in d_1$ and $A \in d_2$. Denote by $M$ the midpoint of $BC$ and by $A'$ the orthogonal projection of $A$ onto $d_1$. Let $P$ be a point on $d_2$ such that $T = PM \cap AA'$ lies in the halfplane bounded by $d_1$ and containing $A$. Prove that there is a point $Q$ on segment $AP$ such that the angle bisector of the angle $BQC$ passes through $T$.

by Nicolae Nica and Cristina Nica, Romania

U94 Let $\Delta$ be the plane domain consisting of all interior and boundary points of a rectangle $ABCD$, whose sides have lengths $a$ and $b$. Define $f : \Delta \to R$, $f(P) = PA + PB + PC + PD$. Find the range of $f$.

by Mircea Becheanu, University of Bucharest, Romania

U123 Let $C_1,C_2,C_3$ be concentric circles with radii $1, 2, 3$,  respectively. Consider a triangle ABC with $A \in C_1, B \in C_2, C \in C_3$. Prove that $max K_{ABC} < 5$, where $max K_{ABC}$ denotes the greatest possible area of triangle ABC.

by Roberto Bosch Cabrera, Havana, Cuba

U237 Let $H$ be a hyperbola with foci $A$ and $B$ and center $O$. Let $P$ be an arbitrary point on $H$ and let the tangent of $H$ through $P$ cut its asymptotes at $M$ and $N$ Prove that $PA + PB = OM + ON$

by Luis Gonzalez, Maracaibo, Venezuela

U239 Let $ABC$ be a triangle and let $P$ be a point in its plane, not lying on the circumcircle $\Gamma$ of triangle $ABC$. Let $AP, BP, CP$ intersect $\Gamma$  again at points $X, Y , Z$, respectively. Let the tangents from $X$ to the incircle of $ABC$ meet $BC$ at $A_1$ and $A_2$, similarly, defi ne $B_1, B_2$ and $C_1, C_2$. Prove that points $A_1, A_2, B_1, B_2, C_1, C_2$ lie on a conic.

by Cosmin Pohoata, Princeton University, USA

U299 Let $ABC$ be a triangle with incircle $\omega$ and let $A', B', C'$ be points outside $\omega$. Tangents from $A'$ to $\omega$ intersect $BC$ at $A_1$ and $A_2$. Points $B_1,B_2$ and $C_1,C_2$ are de ned similarly. Prove that $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a conic if and only if triangle $ABC$ and $A'B'C'$ are perspective.

by Luis Gonzalez, Maracaibo, Venezuela

U310 Let $E$ be an ellipse with foci $F$ and $G$, and let $P$ be a point in its exterior. Let $A$ and $B$ be the points where the tangents from $P$ to $E$ intersect $E$, such that $A$ is closer to $F$. Furthermore, let $X$ be the intersection of $AG$ with $BF$. Prove that $XP$ bisects $\angle AXB$.

by Jishnu Bose, Calcutta, India

U325 Let A1B1C1 be a triangle with circumradius R1. For each n \ge 1, the incircle of triangle AnBnCn is tangent to its sides at points An+1, Bn+1, Cn+1. The circumradius of triangle An+1Bn+1Cn+1, which is also the inradius of triangle AnBnCn is Rn+1. Find lim_n \to +\infty \frac{R_{n+1}}{R_n}.

by Ivan Borsenco, Massachusetts Institute of Technology, USA

U386 Given a convex quadrilateral $ABCD$, denote by $S_A, S_B, S_C, S_D$ the area of triangles $BCD, CDA,DAB,ABC$, respectively. Determine the point $P$ in the plane of the quadrilateral such that $S_A \cdot \overrightarrow{PA} + S_B\cdot \overrightarrow{PB} + S_C\cdot \overrightarrow{PC} + S_D\cdot \overrightarrow{PD} = 0$

by Dorin Andrica, Babes,-Bolyai University, Cluj-Napoca, România

U413 Let $ABC$ be a triangle and let $a,b,c$ be the lengths of sides $BC, CA, AB$, respectively. The tangency points of the incircle with sides $BC, CA, AB$ are denoted by $A', B', C'$.
(a) Prove that the segments of lengths $AA' sinA, BB' sinB, CC' sinC$ are the sides of a triangle.
(b) If $A_1B_1C_1$ is such a triangle, compute in terms of $a, b, c$ the ratio $K_{A_1B_1C_1}/ K_{ABC}$

by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, România

Geometry Articles

Problem Column

2006: Problems & Solutions  problems 001-036
2007: Problems & Solutions  problems 037-072
2008: Problems & Solutions  problems 073-108
2009: Problems & Solutions  problems 109-144
2010: Problems & Solutions  problems 145-180
2011: Problems & Solutions  problems 181-216
2012: Problems & Solutions  problems 217-252
2013: Problems & Solutions  problems 253-288
2014: Problems & Solutions  problems 289-324
2015: Problems & Solutions  problems 325-360
2016: Problems & Solutions  problems 361-396
2017: Problems & Solutions  problems 397-432
2018: Problems & Solutions  problems 433-468
2019: Problems & Solutions  problems 469-504
2020: Problems & Solutions  problems 505-540

sources:
archive.org
reflections.awesomemath.org
www.awesomemath.org

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