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.

232

A SEQUEL TO EUCLID.

half LM. Through L draw QR parallel to PC. Bisect BC in I,
and draw 10 at right angles, and make 210. MN = BI2.
Now, because the Brocard angle is given, the line FE parallel to
the base through the symmedian point is given in position. Hence
QR is given in position; therefore MN is given in magnitude.
Hence IO is given in magnitude; therefore O is a given point.
Again, because F'E is drawn through the symmedian point,
BA2 + AC2: BC2:AM: ML; therefore BI2 + IA2: B2::
MN + NA: MN. Hence BI: IA2:: MN: NA; therefore
IA2 = 210. NA; and since I,,0 are given points, and QR a given
line, the locus of A is a circle coaxal with the point I and the
line QR [VI., Section v., Prop. 1].
6. If on a given line BC, and on the same side of it, be described six triangles equiangular to a given triangle, the vertices
are concyclic.
7. If from the point I, fig., Ex. 5, tangents be drawn to the
Neuberg circle, the intercept between the point of contact and I
is bisected by QR.
8. The Neuberg circles of the vertices of triangles having a
common base are coaxal.
9. In the fig., Prop. 4, p. 175, if the segment A'B' slide along
the line CB', prove that the locus of 0 is a right line.
10. If two triangles be co-symmedians, the sides of one are
proportional to the medians of the other.
11. The six vertices of two co-symmedian triangles form the
vertices of a harmonic hexagon.
12. The angle BOC, fig., Ex. 5, is equal to twice the Brocard
angle of BAC.
13. If the lines joining the vertices of two triangles which
have a common centroid be parallel, their axis of perspective
passes through the centroid. (M'CAY.)
14. The Brocard points of one of two co-symmedian triangles
are also Brocard points of the other.
15. If L, M, N, P, Q, R be the angles of intersection of the
sides of two co-symmedian triangles (omitting the intersections