Lemma
Log in

← All courses

Course · Geometry

Plane geometry I

The full toolkit of the first rounds - congruence, measurement, circles and vectors, in forty problems.

The easiest band of the corpus, drawn from the Serbian municipal round and the Slovenian selection rounds for first-year students, category A, arranged into one path. It starts where every first round starts - two congruent triangles and the one auxiliary point that makes a sum of lengths usable - then learns to compute: exact lengths from right triangles, areas by cutting and subtracting, arcs as fractions of the circle. Circles follow, where tangent segments and the inscribed angle theorem replace measurement by angle chasing, and the course ends with the two ways of solving a geometry problem without a figure at all: vectors, and counting. Ten short lessons name the idea; the problems that follow each one are the real competition papers.

4 chapters · 10 lessons · 40 problems

Lessons and statements are open.Sign in to track progress, open hints, and read solutions.
Sign in

Congruence and auxiliary points

The classical toolkit of the first rounds: read the pair of congruent triangles off the conclusion, build the one point that turns a sum of lengths into a single segment, and halve or double whenever a factor of two appears. Thirteen problems, every one of them decided by triangles you can draw with a ruler.

Congruence from the figureAuxiliary pointHalving devices

  1. LessonSeeing congruenceNext up
  2. Warm-upLet \(ABCD\) be a square and let \(E\) be the midpoint of its side \(CD\). The line through \(D\) perpendicular to the diagonal \(BD\) meets the line \(AE\) at a point \(F\). Prove that the points \(B\), …City
  3. PracticeA triangle is cut into two triangles that are congruent to each other. Prove that the original triangle is isosceles.City
  4. LessonAuxiliary points: turning a sum into a segment
  5. PracticeOn the bisector of the angle \(\angle BAC\) of a triangle \(ABC\), points \(B_1\) and \(C_1\) are chosen so that \(BB_1 \perp AB\) and \(CC_1 \perp AC\). Let \(M\) be the midpoint of the segment \(B_1C_1\). …City
  6. PracticeLet \(CD\) be the bisector of the angle \(BCA\) of a triangle \(ABC\), where \(D\) lies on the side \(AB\), and suppose that \[ AC + BD = BC + AD. \] Prove that the triangle \(ABC\) is isosceles.City
  7. PracticeLet \(ABCD\) be a quadrilateral such that \[ \angle BCA + \angle CAD = 180^{\circ} \qquad\text{and}\qquad AB = AD + BC. \] Prove that \(\angle BAC + \angle ACD = \angle CDA\).City
  8. ConsolidateLet \(ABCDE\) be a convex pentagon whose five sides all have the same length. Suppose that two of its diagonals meet at an angle of \(60^\circ\). Prove that the pentagon has two parallel sides.City
  9. LessonHalving and doubling
  10. PracticeLet \(K\) be the midpoint of the side \(CD\) of a rectangle \(ABCD\). The lines \(BK\) and \(AC\) are perpendicular to each other and meet at the point \(H\), and \(G\) denotes the foot of the perpendicular …City
  11. PracticeLet \(xOy\) be an angle, and let \(A\), \(B\) and \(C\) be points on the arm \(Ox\) such that \(OA = 3\), \(OB = 4\) and \(OC = 6\). Let \(D\) be the foot of the perpendicular dropped from \(B\) to the …City
  12. PracticeIn the right triangle \(ABC\) the right angle is at the vertex \(C\). Let \(S\) be the midpoint of the side \(AB\), and let \(V\) be the point where the altitude dropped from \(C\) meets \(AB\). Determine …City
  13. PracticeIn an isosceles triangle, the bisector of one of the angles at the base is exactly twice as long as the altitude drawn to that base. Determine the angles of the triangle.City
  14. PracticeLet \(M\) and \(P\) be the feet of the perpendiculars from the vertex \(A\) of a triangle \(ABC\) to the bisectors of the exterior angles at the vertices \(B\) and \(C\), respectively. Prove that the length …City
  15. ConsolidateIn a triangle \(ABC\) the angle at \(B\) is obtuse, \(\angle ABC > 90^\circ\), and the side \(AC\) is twice as long as \(AB\), that is \(2\cdot AB = AC\). Prove that \[ 2\cdot\angle ACB > \angle BAC. \] …City
  16. ChallengeIn a triangle \(ABC\) write \(a = BC\), \(b = CA\), \(c = AB\), and let \(S\) be the centre and \(r\) the radius of its inscribed circle. Consider the line joining the midpoints of the sides \(BC\) and …City

Lengths, areas and arcs

The computational chapter of the selection rounds: no shape comes with a formula, every shape comes with a figure. Lengths are trapped in right triangles and released by Pythagoras, one unknown at a time; areas are cut into nameable pieces or obtained by subtracting the piece that has no name; circles contribute sectors, arcs and segments as fractions of the whole. The chapter closes on the two classics of the genre - a lune whose pi terms cancel exactly, and a pizza that comes down to deciding pi against 7 sqrt 3 / 4 by hand.

Metric toolkitAreas and arcs

  1. LessonThe metric toolkit
  2. Warm-upTwo concentric circles of radii \(7\) cm and \(11\) cm are drawn in the plane. The smaller circle cuts a chord of the larger circle into three pieces of equal length. How long is that chord?City
  3. PracticeThe rhombus \(ABCD\) has an acute interior angle at the vertex \(A\). The perpendicular dropped from \(D\) to the side \(AB\) meets it at the point \(E\), which splits the side into the two pieces \[ |AE| = x , \qquad |EB| = y . \] …City
  4. ConsolidateThe figure shows two squares and two congruent circles whose centres lie on a diagonal of the larger square. The smaller square occupies a corner of the larger one, and the diagonal in question runs from …City
  5. LessonAreas without formulas
  6. Warm-upDoes there exist a triangle of area \(1\) whose sides \(b\) and \(c\) satisfy \(c \le b\) and \(b = 1.4\)?City
  7. PracticeA rectangle \(ABCD\) has \(|AB| = 2a\) and \(|AD| = a\). Let \(E\) be the midpoint of the side \(AB\), and choose an arbitrary point \(F\) on the side \(AD\). The area of the triangle \(ECF\) depends on …City
  8. PracticeA rectangle \(ABCD\) satisfies \(|AB| = 10\) and \(|AC| = 5\sqrt{7}\). Let \(M\) be the midpoint of the side \(AB\). Inside the rectangle we draw the semicircle with diameter \(AB\) and the triangle \(CDM\), …City
  9. ConsolidateA circle of radius \(\sqrt{2}\) is given. A second circle, of radius \(2\), has its centre on the first circle. Find the area of the shaded region, that is, of the part of the smaller disc that lies outside …City
  10. ChallengeGrandmother cut a round pizza into \(6\) equilateral triangles and \(6\) circular segments, as in the picture. Each of her \(6\) grandchildren ate one triangle. The grandchildren do not like the crust, …City

Circles: tangents, inscribed angles, concyclicity

The second half of the toolkit. Tangent segments turn length problems into linear algebra; the inscribed angle theorem turns angle conditions into circles you were never given. The chapter ends where the municipal round ends: proving that four points lie on one circle, and using the circle once it is there.

Tangents to a circleInscribed angleConcyclicity

  1. LessonTangents to a circle
  2. Warm-upA circle is inscribed in triangle \(ABC\), whose sides have lengths \(BC = a\), \(CA = b\) and \(AB = c\). A line tangent to this circle meets the side \(AC\) at the point \(P\) and the side \(BC\) at …City
  3. LessonInscribed angles
  4. PracticeIn a quadrilateral \(ABCD\), \[ \angle ABC = 104^\circ, \qquad \angle ADC = 128^\circ, \qquad AB = BC = 2. \] Compute the length of the diagonal \(BD\).City
  5. PracticeProve that in the regular octagon \(A_1A_2A_3A_4A_5A_6A_7A_8\) the diagonals \(A_1A_6\), \(A_3A_7\) and \(A_5A_8\) pass through one point.City
  6. PracticeTwo circles \(k_{1}\) and \(k_{2}\) intersect at two distinct points, and \(AB\) is their common chord. A point \(P\) is chosen on \(k_{1}\) so that it lies outside \(k_{2}\). The lines \(PA\) and \(PB\) …City
  7. LessonProving four points concyclic
  8. PracticeLet \(AA_0\), \(BB_0\) and \(CC_0\) be the altitudes of a triangle \(ABC\), and let \(H\) be its orthocenter. Let \(M\) be the midpoint of the segment \(AA_0\) and let \(N\) be the midpoint of the segment …City
  9. PracticeThree distinct points \(A\), \(B\), \(C\) lie on a line \(\ell\), and a point \(O\) lies off \(\ell\). The perpendicular bisectors of the segments \(OA\), \(OB\) and \(OC\) form a triangle \(EFG\). Prove …City
  10. ConsolidateLet \(S\) be the midpoint of a segment \(AB\), and let \(C\) and \(D\) be points of the semicircle with diameter \(AB\) such that \(C\) lies on the arc \(AD\) and \(\angle CSD = 90^\circ\). Let \(E\) be …City
  11. ConsolidateLet \(ABC\) be a triangle. The tangents to the circumcircle of \(ABC\) at the points \(B\) and \(C\) intersect at a point \(X\). The circle through \(A\), \(B\), \(X\) meets the line \(BC\) again at a …City
  12. ChallengeLet \(X\) be the midpoint of the base \(AB\) of a trapezoid \(ABCD\) (\(AB \parallel CD\)). Prove that if \[ \angle ADX = \angle BCX, \] then the bisectors of the angles \(\angle ADX\), \(\angle DXC\) …City

Vectors, and geometry decided by counting

Two ways of solving a geometry problem without chasing angles. Vectors turn a configuration with too many points into an identity that regroups; parity, angle budgets and the extremal principle settle questions where no two triangles are congruent to begin with. Ten problems, each half of the chapter opening at the bottom of the difficulty band and closing at the top.

Vector languageCounting and extremes

  1. LessonVectors in the plane
  2. Warm-upLet \(T_1\) and \(T_2\) be the centroids of triangles \(A_1B_1C_1\) and \(A_2B_2C_2\), respectively. Prove that \[ \overrightarrow{A_1A_2} + \overrightarrow{B_1B_2} + \overrightarrow{C_1C_2} = 3\,\overrightarrow{T_1T_2}. \] …City
  3. PracticeIn a triangle \(ABC\), let \(C_1\) be the midpoint of the side \(AB\), so that \(CC_1\) is the median from \(C\). Let \(K\) be the midpoint of the segment \(CC_1\), and let the line \(AK\) meet the side …City
  4. PracticeLet \(X\) be a point in the interior of triangle \(ABC\), and let \(T\) be the centroid of that triangle. Points \(M\) and \(N\) lie on side \(BC\), points \(P\) and \(Q\) lie on side \(CA\), and points …City
  5. PracticeLet \(O\) and \(H\) be the circumcenter and the orthocenter of a triangle \(ABC\), and let \(G_1\), \(G_2\), \(G_3\) be the centroids of the triangles \(HBC\), \(HCA\), \(HAB\), respectively. Prove that …City
  6. ConsolidateOne afternoon Ana and Olja each walked in a straight line to visit her boyfriend: Ana to Kosta's house, Olja to Laza's house. The two routes crossed at an old tree, and there the girls met. Standing under …City
  7. ChallengeLet \(ABC\) be a triangle. On side \(AB\) choose points \(C_1\) and \(C_2\) with \[ AC_1 = \tfrac{2015}{3015}\,AB, \qquad AC_2 = \tfrac{2015}{3014}\,AB; \] on side \(BC\) choose points \(A_1\) and \(A_2\) …City
  8. LessonCounting and extremes
  9. PracticeExactly \(2021\) points are chosen on the line \(AB\), and none of them lies on the segment \(AB\). Prove that the sum of the distances from these \(2021\) points to \(A\) can never be equal to the sum …City
  10. PracticeLet \(A\), \(B\), \(C\), \(D\) be four points in the plane, no three of them collinear. Every choice of three of these points forms a triangle, so the four points determine \(12\) angles in all. Write …City
  11. PracticeCan an equilateral triangle be divided - that is, actually cut up with scissors - into \(2006\) equilateral triangles?City
  12. ChallengeIs it possible to divide a square into convex pentagons?City

Problems from the municipal round (opstinsko takmicenje) of the Serbian mathematical competition, high school grade I, category A, 2000-2026, organized by the Mathematical Society of Serbia (DMS), and from the Slovenian selection rounds (izbirno/odbirno tekmovanje, 1. letnik, kategorija A, 2001-2024) organized by DMFA Slovenije. Statements and solutions are re-expressed in English; the mathematical content follows the official papers.