This page is dedicated to the memory of Nicolae Teodorescu (1908-2000), for all his encouragments, since I was just thirteen...
Assigned at the final exam of the national contest of the monthly Gazeta Matematicã in August 1989, Câmpulung-Muscel, Romania.
The question (a) is coming from F.G.-M.
1. Let XY be a secant intersecting the circle C(O,R) of center O
and radius R. The midpoint of the small arc XY is denoted T . An arbitrary
line passing through T intersects the large arc XY in A. The projection of T
on XY is K, of X on AT is H, of Y on AT is L and,
finally, the projection of A on XY is G. Prove that:
(a) The points G, H, K, L are on the same circle;
(b) KH and AY are parallel;
(c) If E is the midpoint of AX, then the center of the circle determined by G, H,
K and L lies on the circumcircle of the triangle GEK.
Remark : The problem was proposed to the IXth and the Xth grades section. In fact, 52 % of the contest participants solved it completely. Out of all the students in the contest, 16 % have solved or tried to solve it by using metric relations between segments. The others have used relationships between angles.
The problem was included in author's longer paper Methods and Structures in Plane Geometry , published in five series in the Romanian monthly Preuniversitaria. This problem appeared in part III, in No. 48, 1990.
A joint problem with Marcel Tena :
2. Describe a class of triangles ABC with the property that tan A, tan B and tan C are integers.
This problem was assigned at the final exam of the national contest of the monthly Gazeta Matematicã, Mangalia, Romania, September 1989.
Variation on a theme from P. Aubert and G. Papelier, Exercices de géométrie analitique , Vuibert, Paris, 1930.
3. Consider the circle O and the fixed points A and B. Let us consider a line through A intersecting the circle in C and D. Connect B with the midpoint E of CD. Find the geometric loci of the midpoint of BC, when the line AC is variable. Discuss all the cases.
This problem was never assigned during a test or proposed in a review. It was used by author during his recitations for the class of Methods in Teaching Mathematics, Fall term 1995 at University of Bucharest. It was also used in Methods and Structures in Plane Geometry, part III, Preuniversitaria, No.48, 1990.
A problem assigned during the final exam of the national contest of the monthly Gazeta Matematicã, Mangalia, Romania, September 1989.
4.Let T the projection of the orthocenter H of the acute triangle ABC on the bisecting line of the angle A. The intersection of this bisecting line with BC is
called A'. Denote by P the projection of T on BC, M the midpoint of
BC, M the midpoint of BC.
(a) Prove that TA' is the bisecting line of the angle MTP.
(b) Prove that TM is parallel to AO, where O is the center of the circumcircle of the triangle ABC.
Remark : In the contest, the problem got four distinct solutions from students. The author's solution was published in his paper Methods and Structures in Plane Geometry, part V, Preuniversitaria, No.50, 1991.
The problem No. L.51 from Computer Matematica, issue 2(6)/1995:
5. On the sides AB, BC, CA of the triangle ABC consider the points C', A', M such that MA' = MC and MC'=MA. Find the geometric locus of the center of the circumcircle of the triangle A'BC'.
The problem was published with the remark: Related to the problem 3516 from Matematika v shkole, issue 6/1990.
A problem published also in Computer Matematica, issue 1(4)/1994, for the seventh elementary grade. The problem number is G.43.
6.In the triangle ABC we divide each of the angles B and C in three equal parts. Let the rays d' and d" be the trisectors of the angle B and e' and e" be the trisectors of the angle C. (The notation counts the rays from BC toward AB, respectively AC.) Let us denote by N the intersection of d' and e' and by M the intersection of d" and e". Prove that MN is perpendicular on BC if and only if the angle B has the same measure as the angle C.