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.
26. If a semicircle and its diameter be touched by any circle, either internally or externally, twice the rectangle contained by the radius of the semicircle, and the radius of the tangential circle, is equal to the rectangle contained by the segments of any secant to the semicircle, through the point of contact of the diameter and touching circle.