25. The line joining the middle points of the diagonals of a quadrilateral circumscribed to a circle—
- divides each pair of opposite sides into inversely proportional segments;
- is divided by each pair of opposite lines into segments which, measured from the centre, are proportional to the sides;
- is divided by both pairs of opposite sides into segments which, measured from either diagonal, have the same ratio to each other.
26. If CD, CD′ be the internal and external bisectors of the angle C of the triangle ACB, the three rectangles AD.DB, AC.CB, AD.BD′ are proportional to the squares of AD, AC, AD′; and are—(1) in arithmetical progression if the difference of the base angles be equal to a right angle; (2) in geometrical progression if one base angle be right; (3) in harmonical progression if the sum of the base angles be equal to a right angle.
27. If a variable circle touch two fixed circles, the chord of contact passes through a fixed point on the line connecting the centres of the fixed circles.
Dem.—Let O, O′ be the centres of the two fixed circles; O′′ the centre of the variable circle; A, B the points of contact. Let AB and OO′ meet in C, and cut the fixed circles again in the points A′, B′ respectively. Join A′O, AO, BO′. Then AO, BO′ meet in O′′ [III. xi.]. Now, because the triangles OAA′, O′′AB are isosceles, the angle O′′BA = O′′AB = OA′A. Hence OA′ is parallel to O′B; therefore OC : O′C :: OA′ : O′B; that is, in a given ratio. Hence C is a given point.
28. If DD′ be the common tangent to the two circles, DD′2 = AB′.A′B.
29. If R denote the radius of O′′ and ρ, ρ′, the radii of O, O′, DD′2 : AB2 :: (R±ρ)(R±ρ′) : R2, the choice of sign depending on the nature of the contacts. This follows from 28.
30. If four circles be tangential to a fifth, and if we denote by 12 the common tangent to the first and second, &c., then
