10. On the sides AB, AC of a triangle ABC, equilateral triangles
ABF, ACE are drawn outside the triangle ABC. N is the
centroid of the triangle ABF and M is the centroid of the
triangle ACE.
By applying the cosine formula to the triangle AMN, or
otherwise, prove that
MN2 = -
6
62 + c2 + a2
abc'\/3\
R J
where R is the radius of the circumcircle of triangle ABC.
If the equilateral triangle BCD is drawn outwards on BC and
L is the centroid of this triangle, show that the triangle
LMN is equilateral.
11. A variable chord CD of a circle is perpendicular to a fixed
diameter AB and meets AB at P. If AP = x, show that the
area of the triangle ACD is given by Vx3 (d — x), where d
is the diameter of the circle.
Find the position of P for which this area is a maximum.
Flence, or otherwise, determine the shape of the triangle of
maximum area which can be inscribed in a given circle.
(C41644)
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