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Course · Geometry

Plane geometry II

Similarity, circles at work, and the triangle's centres - the geometry of the regional rounds.

The middle band of the corpus, drawn from the Serbian okruzno round and the Slovenian regijsko and neighbouring rounds for first-year students, category A. It opens with the first genuinely new tool after congruence - similar triangles, the intercept theorem, and the square law that ties a ratio of lengths to a ratio of areas - then puts the circle machinery of Plane geometry I to work on regional-round configurations, and closes with the triangle's centres: the incentre and its tangent lengths, the circumcentre and its central angles, the orthocentre and its reflections, all the way to the Euler line. Two lessons introduce the new theory; everything else is taught by the problems themselves, each cluster opening with a warm-up.

3 chapters · 2 lessons · 33 problems

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Similar triangles and ratios

The first genuinely new tool after congruence: triangles of the same shape at different scales. One matching pair of angles buys a whole proportion of sides; a triangle similar to a piece of itself turns a shared side into a geometric mean; and the square of the ratio carries similarity from lengths to areas. The chapter ends the way regional rounds like to end: comparing areas that are never actually computed.

Similar trianglesGeometric meanArea ratios

  1. LessonSimilar trianglesNext up
  2. Warm-upLet \(ABCD\) be a quadrilateral and let \(K\) be a point inside triangle \(ABD\) such that triangles \(ABD\) and \(KCD\) are similar, the vertices corresponding in the written order. Prove that triangles …City
  3. PracticeLet \(ABC\) be an isosceles triangle with apex \(C\). Points \(D\) and \(E\) lie on the sides \(AC\) and \(BC\) respectively, and the bisector of the angle \(\angle DEB\) and the bisector of the angle …City
  4. ConsolidateThe horizontal segment in the picture is divided into six parts of equal length, and every triangle appearing in the picture is equilateral. The whole figure is shaded in two colours, light grey and dark …National
  5. Warm-upThe lengths of the sides of a triangle are three consecutive natural numbers, each greater than \(3\). The altitude drawn to the middle side splits that side into two segments. Prove that the lengths of …Regional
  6. PracticeIn a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at the point \(D\). The triangle \(ADC\) is isosceles with apex \(D\), that is, \(|DA| = |DC|\). Given \(|CD| = 36\) …Regional
  7. ConsolidateThree straight cuts divide a rectangle into four pieces, as shown in the figure; the cut meeting the top edge is perpendicular to it. The four pieces are then rearranged, without gaps or overlaps, into …Regional
  8. Warm-upEach diagonal of a quadrilateral \(ABCD\) divides it into two parts of equal area. Prove that \(ABCD\) is a parallelogram.Regional
  9. PracticeJanez drew a pattern on a sheet of paper, made up of congruent squares and congruent hexagons. On top of the pattern he then drew two dashed lines perpendicular to each other, as in the figure. What is …Regional
  10. PracticeIn a convex hexagon \(ABCDEF\), each of the two diagonals \(AD\) and \(BE\) divides the hexagon into two pieces of equal area. Prove that the quadrilateral \(BDEA\) is a trapezoid.Republic
  11. ConsolidateLet \(t_a\) and \(t_b\) be the medians of a triangle \(ABC\) drawn to the sides \(BC\) and \(CA\), and let \(P\) be the area of the triangle. Prove that \[ t_a \cdot t_b \geqslant \tfrac{3}{2} P , \] and …Regional
  12. ChallengeLet \(ABC\) be a right triangle with its right angle at \(C\), and write \(|BC| = a\), \(|AC| = b\). Let \(D\) be a point on the opposite side of the line \(AC\) from \(B\) for which the triangle \(ACD\) …Regional

Circles at work

No new theorems: the tangents, inscribed angles and concyclicity tests of Plane geometry I now have to carry regional-round arguments on their own. Tangents supply perpendicular radii and isosceles triangles, a cyclic quadrilateral moves an angle to the one place where it can be compared, and an equal pair of arcs betrays the reflection or rotation that finishes the proof. Each concept opens with a warm-up problem instead of a lesson - the theory is known, the length of the argument is what is being trained.

Tangent configurationsCyclic quadrilaterals at workEqual arcs and symmetry

  1. Warm-upTwo circles touch each other internally at a point \(A\). Let \(AB\) be a diameter of the larger circle. Through the other endpoint \(B\) of this diameter a line is drawn which touches the smaller circle …Regional
  2. PracticeIn the plane, two circles \(k_1\) and \(k_2\) and a line \(p\) are given. The line \(p\) cuts \(k_1\) at the points \(A\) and \(B\), and it cuts \(k_2\) at the points \(C\) and \(D\). Each of the two tangents …Regional
  3. ConsolidateLet \(ABC\) be an acute triangle. The circle \(k\) with diameter \(AB\) meets the side \(AC\) at \(M\) and the side \(BC\) at \(N\). The tangents to \(k\) at \(M\) and at \(N\) meet at the point \(P\). …Regional
  4. ConsolidateIn triangle \(ABC\) the angle at \(A\) is a right angle. Points \(D\), \(E\) and \(F\) are chosen on the sides \(AB\), \(BC\) and \(CA\) respectively, so that \[ |BD| = |BE| \qquad\text{and}\qquad |CF| = |CE| , \] …National
  5. Warm-upFind all points \(P\) on the circle circumscribed about a triangle \(ABC\) for which the sum \[ PA + PB + PC \] is as small as possible.Regional
  6. PracticeTwo equilateral triangles \(ABC\) and \(PQR\) lie in the plane so that \(R\) is an interior point of the segment \(AB\) and \(C\) is an interior point of the segment \(PQ\), the points \(A\) and \(P\) …Regional
  7. PracticeThe points \(A, B, C, D, E\) lie on one circle in such a way that \(A\) and \(D\) are on opposite sides of the line \(BC\), and \(B\) and \(E\) are on opposite sides of the line \(CD\). Given that \[ \angle ABC = \angle BCD = \angle CDE = 45^\circ , \] …Regional
  8. Warm-upOn the sides \(AB\) and \(BC\) of an equilateral triangle \(ABC\), points \(Z\) and \(X\) are chosen so that \[ AZ : ZB = BX : XC = 2021 : 2020. \] The perpendicular bisector of the segment \(XZ\) meets …Regional
  9. PracticeCircles \(k_1\) and \(k_2\) intersect at points \(P\) and \(Q\), and \(k_1\) passes through the centre of \(k_2\). Distinct points \(A\) and \(B\) lie on the arc of \(k_1\) that runs inside \(k_2\), and …Regional
  10. ConsolidateCongruent circles \(k_1\), \(k_2\) and \(k\) all pass through a point \(P\), and each pair of them meets in one further point: \(k\) and \(k_1\) meet again at \(A\), \(k\) and \(k_2\) meet again at \(B\), …Regional
  11. ChallengeOn the sides of a triangle \(ABC\), equilateral triangles \(ADB\), \(BEC\) and \(CFA\) are constructed outwardly, so that \(D\), \(E\), \(F\) are the apexes over \(AB\), \(BC\), \(CA\) respectively. Prove …Regional

The centres of the triangle

The finale of the course: the incentre, the circumcentre and the orthocentre as characters with fixed habits. The incentre sees a side under 90 degrees plus half the opposite angle, the circumcentre's direction is read off a central angle, and the orthocentre - almost never named in a statement - is summoned because its reflections land on the circumcircle and its vector from the circumcentre is the sum of the three vertex vectors. Eleven regional-round problems, each decided by naming the right centre and quoting the right habit.

IncentreCircumcentreOrthocentre configurations

  1. LessonThe centres of the triangle
  2. Warm-upThe incircle of a triangle \(ABC\) touches the sides \(AB\) and \(AC\) at the points \(D\) and \(E\) respectively. Let \(F\) be any point of the side \(AB\) lying between \(A\) and \(D\), and let \(G\) …City
  3. PracticeIn triangle \(ABC\) the bisector of the angle \(CAB\) meets the side \(BC\) at the point \(N\), and the bisector of the angle \(CBA\) meets the side \(AC\) at the point \(P\), where \[ PN = a . \] Let …Regional
  4. PracticeIn a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at \(D\), and the bisector of the angle \(\angle CBA\) meets the side \(AC\) at \(E\). Suppose that \(|CD| = |CE|\). …Regional
  5. ConsolidateThe circle inscribed in triangle \(ABC\) touches the sides \(BC\), \(CA\) and \(AB\) at the points \(D\), \(E\) and \(F\) respectively. A point \(K\) lies on the same side of the line \(EF\) as the vertex …Regional
  6. Warm-upLet \(H\) be the orthocenter of a triangle \(ABC\), and let \(K\) be the point symmetric to \(H\) with respect to the midpoint of the side \(BC\). Prove that \(AK\) is a diameter of the circumcircle of …Regional
  7. PracticeThe line through the circumcentre and the orthocentre of a triangle \(ABC\) (the Euler line of the triangle) crosses the interior of the side \(CA\) at a point \(M\) and the interior of the side \(CB\) …Regional
  8. PracticeIn a triangle \(ABC\) the bisector of the angle at the vertex \(A\) meets the side \(BC\) at the point \(D\). The perpendicular dropped from \(B\) to the line \(AD\) meets the circumcircle of the triangle …Regional
  9. ConsolidateLet \(T\) be the centroid of an acute triangle \(ABC\). Let \(A'\) be the foot of the altitude drawn from \(A\) to the side \(BC\), and let \(A''\) be the point of the segment \(BC\) for which \[ BA' = A''C . \] …Regional
  10. Warm-upOn the sides of an acute triangle \(ABC\) points \(A_1 \in BC\), \(B_1 \in CA\) and \(C_1 \in AB\) are chosen so that \[ \angle CC_1B = \angle AA_1C = \angle BB_1A = \varphi, \] where \(\varphi\) is an …Regional
  11. ConsolidateLet \(H\) be the orthocentre of an acute triangle \(ABC\), and let \(A_1\), \(B_1\) and \(C_1\) be the centres of the circles circumscribed about the triangles \(BHC\), \(CHA\) and \(AHB\) respectively. …Regional
  12. ChallengeLet \(ABC\) be a triangle with \(BC \neq CA\), and let \(H\), \(T\) and \(O\) be its orthocentre, centroid and circumcentre. Let \(P\) be the point symmetric to \(T\) with respect to \(O\), and let \(Q\) …Regional

Problems from the Serbian okruzno and republicko rounds, high school grade I, category A, 1995-2023, organized by the Mathematical Society of Serbia (DMS), and from the Slovenian mathematical competitions (izbirno, regijsko and drzavno rounds, 1. letnik, kategorija A, 2006-2023) organized by DMFA Slovenije. Statements and solutions are re-expressed in English; the mathematical content follows the official papers.