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. 19. If R denotes the radius of the circle circumscribed about a triangle ABC, r, r′, r′′, r′′′ the radii of its inscribed and escribed circles, δ, δ′, δ′′ the perpendiculars from its circumcentre on the sides; μ, μ′, μ′′ the segments of these perpendiculars between the sides and circumference of the circumscribed circle, we have the relations— r′ + r′′ + r′′′ = 4R + r, (1) μ + μ′ + μ′′ = 2R − r, (2) δ + δ′ + δ′′ = R + r. (3) The relation (3) supposes that the circumcentre is inside the triangle. 20. Through a point D, taken on the side BC of a triangle ABC, is drawn a transversal EDF, and circles described about the triangles DBF, ECD. The locus of their second point of intersection is a circle. 21. In every quadrilateral circumscribed about a circle, the middle points of its diagonals and the centre of the circle are collinear. 22. Find on a given line a point P, the sum or difference of whose distances from two given points may be given. 23. Find a point such that, if perpendiculars be let fall from it on four given lines, their feet may be collinear. 24. The line joining the orthocentre of a triangle to any point P, in the circumference of its circumscribed circle, is bisected by the line of collinearity of perpendiculars from P on the sides of the triangle. 25. The orthocentres of the four triangles formed by any four lines are collinear.