Problem-Solving with GeoGebra
József Kosztolányi
1. The equation of the circumcircle of symmetric trapezoid ABCD is x  32   y  22  100 .
The equation of the axis of the trapezoid is 2 x  y  4 and P(−5; 1) is an inner point of the
base AB. Find the coordinates of the vertices of the trapezoid, if BC  10 2 .
2. A central symmetric convex hexagon ABCDEF is given. We have P on side BC, Q on side
DE, and R on side FA are arbitrary points. Prove that the area of triangle PQR is at most half
of the area of hexagon.
3. Discover and prove the Ceva’s Theorem: In triangle ABC we have C1 on side AB, A1 on
side BC, and B1 on side CA. The lines AA1, BB1, and CC1 intersect at a common point if and
only if
AC1 BA1 CB1


1.
C1 B A1 B B1 A
4. We have function f : R  R, f x   x 2  4 x  a , where a   5; 5 is a parameter. Find the
area (depend on a) determined by the graphs of f, and f  .
Problem-Solving with GeoGebra
József Kosztolányi
1. The equation of the circumcircle of symmetric trapezoid ABCD is x  32   y  22  100 .
The equation of the axis of the trapezoid is 2 x  y  4 and P(−5; 1) is an inner point of the
base AB. Find the coordinates of the vertices of the trapezoid, if BC  10 2 .
2. A central symmetric convex hexagon ABCDEF is given. We have P on side BC, Q on side
DE, and R on side FA are arbitrary points. Prove that the area of triangle PQR is at most half
of the area of hexagon.
3. Discover and prove the Ceva’s Theorem: In triangle ABC we have C1 on side AB, A1 on
side BC, and B1 on side CA. The lines AA1, BB1, and CC1 intersect at a common point if and
only if
AC1 BA1 CB1


1.
C1 B A1 B B1 A
4. We have function f : R  R, f x   x 2  4 x  a , where a   5; 5 is a parameter. Find the
area (depend on a) determined by the graphs of f, and f  .