Def.—The circle round O as centre, the square of whose radius is equal OA.OP = OB.OQ = OC.OR, is called the polar circle of the triangle ABC.
Observation.—If the orthocentre of the triangle ABC be within the triangle, the rectangles OA.OP, OB.OQ, OC.OR are negative, because the lines OA.OP, &c., are measured in opposite directions, and have contrary signs; hence the polar circle is imaginary; but it is real when the point O is without the triangle—that is, when the triangle has an obtuse angle.
3. If the perpendiculars of a triangle be produced to meet the circumscribed circle, the intercepts between the orthocentre and the circle are bisected by the sides of the triangle.
4. The point of bisection (I) of the line (OP) joining the orthocentre (O) to the circumference (P) of any triangle is equally distant from the feet of the perpendiculars, from the middle points of the sides, and from the middle points of the distances of the vertices from the orthocentre.
Dem.—Draw the perpendicular PH; then, since OF, PH are perpendiculars on AB, and OP is bisected in I, it is easy to see that IH = IF. Again, since OP, OG are bisected in I, F; IF =
PG—that is, IF =
the radius. Hence the distance of I from the foot of each perpendicular, and from the middle point of each side, is =