8. In the same case, if K be the point of contact with BC of the escribed circle, which touches the other sides produced, LH.BK = BD.LE.
9. If R, r, r′, r′′, r′′′ be the radii of the circumscribed, the inscribed, and the escribed circles of a plane triangle, d, d′, d′′, d′′′ the distances of the centre of the circumscribed circle from the centres of the others, then R2 = d2 + 2Rr = d′2 − 2Rr′, &c.
10. In the same case, 12R2 = d2 + d′2 + d′′2 + d′′′2.
11. If p′, p′′, p′′′ denote the perpendiculars of a triangle, then
| (1) | + + = ; | ||
| (2) | + − = , &c.; | ||
| (3) | = −, &c.; | ||
| (4) | = + , &c. |
12. In a given triangle inscribe another of given form, and having one of its angles at a given point in one of the sides of the original triangle.
13. If a triangle of given form move so that its three sides pass through three fixed points, the locus of any point in its plane is a circle.
14. The angle A and the area of a triangle ABC are given in magnitude: if the point A be fixed in position, and the point B move along a fixed line or circle, the locus of the point C is a circle.
15. One of the vertices of a triangle of given form remains fixed; the locus of another is a right line or circle; find the locus of the third.
16. Find the area of a triangle—(1) in terms of its medians; (2) in terms of its perpendiculars.