A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

246

A SEQUEL TO EUCLID.

130. If two circles, W, W', coaxal with the circumcircle and
Brocard circle of a harmonic polygon, be inverse to each other,
with respect to the circumcircle; then the inverses of the circumcircle and the circle W, with respect to any point in the circumference of W', are respectively the circumcircle and Brocard circle
of another harmonic polygon whose vertices are the inverses of
the vertices of the former polygon.
131. If R' be the radius of the cosine circle of a harmonic
polygon of n sides; A, a, the diameters of its Lemoine circle and
Brocard circle, respectively; then
7r
A2 - 8 = R'2 se2 -.
132. If the vertices of a harmonic polygon of n sides be inverted from any arbitrary point into the vertices of another harmonic polygon, the inverses of the centres of inversion of the
former will be the centres of inversion of the latter.
133. The mean centre of the vertices of a cyclic quadrilateral
is a point in the circumference of the nine-point circle of the
harmonic triangle of the quadrilateral. (RUSSELL.)
134. Prove that in the plane of any triangle there exist two
points whose pedal triangles with respect to the given triangle
are equilaterals.
135. Prove that the loci of the centres of the circumcircles of
the figures F2, F3, Ex. 13, page 222, are circles.
136. If A', B', C' be the points where Malfatti's circles touch
each other, prove that the triangles ABC, A'B'C' are in perspective.
137. Prove the following construction for Steiner's point R
(Ex. 80). With the verticesA, B, C of the triangle as centres, and
with radii equal to the opposite sides, respectively, describe
circles. These, it is easy to see, will intersect, two by two,
on the circumcircle in points A,, B1, C1. Then the line joining the
intersection of BC and B1C1 to A, will meet the semicircle in the
point required.
138-140. If the perpendiculars of the triangle ABC, produced
if necessary, meet the circles of Ex. 137 in the points A', B', C',
prove1~. Area of A A'B'C' = 4 cot w. A ABC.
2~. Sum of squares of A'B', B'C', C'A'
= 8 A ABC (2 cot w - 3).
3~. If w' be the Brocard angle of A'B'C',
cot w' = (2 cot c - 3) / (2 - cot o). (NEUBERG.)