Three equilateral triangles ABC, AYX and XZB are drawn with the
point X a moveable point on AB. The points P, Q and R are the
centres of the three triangles. What can you say about triangle
PQR?
P is the midpoint of an edge of a cube and Q divides another edge
in the ratio 1 to 4. Find the ratio of the volumes of the two
pieces of the cube cut by a plane through PQ and a vertex.
Halving the Triangle
Age 16 to 18 Challenge Level
Draw any triangle P, Q, R.
Find points A, B and C, one on each side of the triangle (but
not at the vertices), such that the area of triangle ABC is one
quarter the area of triangle PQR.
Now find points such that the area of triangle ABC is one half
the area of triangle PQR and a method that will give any fraction
of the area between one quarter and one.
How would you adapt your method to give fractions less than one
quarter?