If the point be within the triangle, x, y, z its perpendicular distances from the sides, we at once deduce that 22+ y2+z2 is a nin. when

It may be noticed that the point thus determined is the 'Symmedian' point. See Paper LXVIII., 4.

we see that the middle point of the base of a triangle is such that the product of the perpendiculars from it to the sides is a min.; and we deduce that the point within a triangle such that the product of the perpendicular distances from it to the sides is a min. is the centroid of

the triangle, and that the value of the min. product is

43 abc

For further treatment of this subject, see an elementary treatise by the present author on Geometrical and Algebraical Max. and Min. without the aid of Differential Calculus, where it is shewn that the problem of finding the max. or min. values of geometrical magnitudes depending only on the point line and circle is reduced to the much simpler one of finding its positions of symmetry.

THE END.

RICHARD CLAY & SONS,

BREAD STREET HILL, LONDON, E.C and Bungay, Suffolk.

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