The following questions were used at a recent examination:
1. Theorem: The three medians of any triangle intersect in a common point which is at two-thirds of the distance from each vertex to the middle of the opposite side.
2. Theorem: If two triangles have their three sides respectively equal, the triangles are equal in all respects.
3. (a) How many circles can be drawn tangent to three given straight lines? (b) Problem: To draw a circle through a given point and tangent to two given straight lines.
4. Theorem: If two parallel right lines be divided into corresponding parts, proportional each to each, and straight lines be drawn through the corresponding points of division, these straight lines will pass through a common point.
5. Exercise: Find the locus of all points, the sum of the squares of the distances of any one of which from two fixed points is equal to a given square.
6. Problem: Given two circles, to construct a third circle equivalent to their difference.
7. Exercise: If the radius of a circle is 5, find the area of the segment subtended by the side of a regular hexagon.
8. Theorem: The areas of two triangles which have an angle of the one equal to an angle of the other, are to each other as the products of the sides including those angles.
9. Problem: Through a given point on one side of a triangle to draw a right line which shall divide the triangle into two equivalent areas.