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''.

Geometry problems with aops links
(2006 issues)

collected inside aops here:

Junior level                        

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.

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

S2  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 $OI$.

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.

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.

by Mihai Miculita, Oradea, Romania

Olympiad level

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.

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

Geometry Articles

  1. A Characterization of the Parallelogram, by Paris Pamfilos 2016.3
  2. A Fine Use of Transformations, by Rithvik Pasumarty 2016.4
  3. A Forgotten Coaxality Lemma, by Stanisor Stefan Dan 2015.5
  4. A Generalization of the Napoleon’s Theorem, by Khakimboy Egamberganov 2017.3
  5. A Metric Relation and its Applications, by Son Hong Ta 2008.2
  6. A minimum problem, by Laurentiu Panaitopol 2006.6
  7. A new proof for Napoleon's Theorem, by Alex Anderson  2007.3
  8. A Note on Power of a Point, by Michal Rolinek & Josef Tkadlec 2010.5
  9. A Nice Theorem on Mixtilinear Incircles, by Khakimboy Egamberganov 2016.4
  10. A Note on the Malfatti Problem, by Titu Andreescu & Oleg Mushkarov 2006.4
  11. A Short Proof of Lamoen's Generalization of the Droz-Farny Line Theorem, by Cosmin Pohoata & Son Hong Ta 2011
  12. A Special Point on the Median, by Anant Mudgal & Gunmay Handa 2017.2
  13. A Way to Prove a Geometric Inequality R $\ge$ 3r, by Nguyen Tien Lam 2009.1
  14. All About Excircles!, by Prasanna Ramakrishnan 2014.6
  15. Angle Inequalities in Tetrahedra, by Mark Chen 2008.6
  16. Back to Euclidean Geometry: Droz-Farny Demystified, by Titu Andreescu & Cosmin Pohoata 2012
  17. Concyclicities in Tucker-like Configurations, by Stefan Dominte & Tudor-Dimitrie Popescu 2016.1
  18. Droz-Farny, an Inverse View, by Paris Pamfilos 2015.2
  19. Equiangular polygons: An algebraic approach, by Titu  Andreescu & Bogdan Enescu 2006.1
  20. Harmonic Division and its Applications, by Cosmin Pohoata 2007.4
  21. Inequalities on Ratios of Radii of Tangent Circles, by Y.N. Aliyev 2015.5
  22. From Baltic Way to Feuerbach - A Geometrical Excursion, by Darij Grinberg 2006.2
  23. Hartcourt’s Theorem Via Salmon’s Lemma, by Luis Gonzalez and Cosmin Pohoata 2013.6
  24. How to make a soccer ball, Euler's relation for polyhedra and planar graphs, by Bogdan Enescu 2013.4
  25. Let's Talk About Symmedians! , by Sammy Luo and Cosmin Pohoata 2013.4
  26. Newton and Midpoints of Diagonals of Circumscriptible Quadrilaterals, by Titu  Andreescu, Luis Gonzalez & Cosmin Pohoata 2014.1
  27. On a Mixtilinear Coaxality, by Cosmin Pohoata & Vladimir Zajic 2012
  28. On a vector equality, by Nguyen Tien Lam 2008.6
  29. On Casey's Inequality, by Tran Quang Hung 2011.2
  30. On Distances In Regular Polygons, by N. Javier Buitrago A. 2010.5
  31. On Fontene’s Theorems, by Marius Bocanu  2015.2
  32. On some geometric inequalities, by Tran Quang Hung 2008.3
  33. On the area of a pedal triangle, by Ivan Borsenco 2007.2
  34. On the extension of Carnot's Theorem, by Tran Quang Hung 2007.6
  35. On the Maximum Area of a Triangle With the Fixed Distances From its Vertices to a Given Point , by Ivan Borsenco, Roberto Bosch Cabrera & Daniel Lasaosa 2014.2
  36. On vector properties of an equilateral triangle, by Tran Quang Hung 2007.2
  37. Similar Quadrilaterals, by Guangqi Cui, Akshaj Kadaveru, Joshua Lee, Sagar Maheshwari 2014.5
  38. The Apollonian Circles and Isodynamic Points, by Tarik Adnan Moon 2010.6
  39. The Monge-D’Alembert Circle Theorem, by Cosmin Pohoata & Jan Vonk 2011.5
  40. The Neuberg-Mineur circle, by Darij Grinberg
  41. The Symmedian Point and Poncelet’s Porism
  42. Triangle Bordered With Squares, by Catalin Barbu 2010.4
  43. Triangles Homothetic with the Intouch Triangle, by Sava Grozdev, Hiroshi Okumura, & Deko Dekov 2018.2
  44. Vectors Conquering Hexagons, by Iurie Boreico 2008.1

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 469-474)
1: Problems Solutions problems 469-474
2: Problems & Solutions problems 475-480
3: Problems & Solutions  problems 481-495


1 σχόλιο: