This historic book may have numerous typos and missing text.
Purchasers can download a free scanned copy of the original book
(without typos) from the publisher. Not indexed. Not illustrated.
1891 Excerpt: ...line of a triangle and theorems connected with it.
By George A. Gibson, M.A Most of the following theorems occur in a
more or less explicit form in text-books on the geometry of the
parabola; but it will not, I hope, be without interest and value to
consider them independently, and to prove them by using only the
propositions of Euclid. 1. If the perpendiculars AD, BE, CF of a
triangle ABC are produced to meet the circumcircle of the triangle
in X, Y, Z respec tively; and if through the orthocentre 0 any line
be drawn meeting the sides BC, OA, AB in U, V, W respectively, then
XU, YV, ZW intersect the circumcircle at the same point S, the
pedal of which is parallel to the line through O and is midway
between S and that line. From the orthocentre O (fig. 26) draw any
line OU cutting BC at XJ and let XU meet the circumcircle at S.
Draw SL perpendicular to BC, meeting OU at P. Assume the property
that OD is equal to DX. Then since the triangle ODU, XDU are
congruent, so are the triangles SLU, PLU and therefore SL = LP.
Also UOX= uxo= SCA. Similarly it may be shown that if SY meet CA in
V and if SM be drawn perpendicular to CA and produced to meet OV at
Q, then SM = MQand z.VOY=SCB. Hence Z.UOX+ Lvoy = L ACB =
supplement of Z.XOY. Therefore OU and OV are in the same straight
line. Similarly it may be shown that ZS meets AB at W, the point of
intersection of UO and AB, and that if SN be drawn perpendicular to
AB and produced to meet UW at R, then SN = NR. L, M, N being the
middle points of SP, SQ, SR respectively, are therefore collinear,
and obviously LM bisects SO. 2. If on the pedal line a point G be
taken (fig. 27) and if through G a line be drawn perpendicular to
SG meeting the sides BC, CA, AB in A', B', C respectively, then
SA', SB', SC make equal angles w...
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