10. If the perpendicular to any side of a triangle at its middle point meet the internal and external bisectors of the opposite angle in the points D and E; prove that D, E are points on the circumscribed circle.
11. Through a given point P draw a chord of a circle so that the intercept EF may subtend a given angle X.
12. In a given circle inscribe a triangle having two sides passing through two given points, and the third parallel to a given line.
13. Given four points, no three of which are collinear; describe a circle which shall be equidistant from them.
14. In a given circle inscribe a triangle whose three sides shall pass through three given points.
15. Construct a triangle, being given—
- The radius of the inscribed circle, the vertical angle, and the perpendicular from the vertical angle on the base.
- The base, the sum or difference of the other sides, and the radius of the inscribed circle, or of one of the escribed circles.
- The centres of the escribed circles.
16. If F be the middle point of the base of a triangle, DE the diameter of the circumscribed circle which passes through F, and L the point where a parallel to the base through the vertex meets DE: prove DL.FE is equal to the square of half the sum, and DF.LE equal to the square of half the difference of the two remaining sides.
17. If from any point within a regular polygon of n sides perpendiculars be let fall on the sides, their sum is equal to n times the radius of the inscribed circle.
18. The sum of the perpendiculars let fall from the angular points of a regular polygon of n sides on any line is equal to n times the perpendicular from the centre of the polygon on the same line.