95 
20. If the three perpendiculars from the vertices of one triangle upon the 
sides of another triangle are concurrent, then the three perpendiculars from the 
vertices of the latter upon the sides of the former are also concurrent. (Steiner, 
Gesammelte Werke I[., 157, 1881.) (Lemoine calls such triangles orthologous and 
the points of concurrency centers of orthology. 
21. Brocard’s Triangle and ABC are orthologous. Perpendiculars from 
A,, B,, C,, upon BC, CA, AB, respectively, are concurrent. (See No. 13.) 
22. If three points be taken on the sides of a triangle such that the sums of 
the squares of the alternate segments taken cyclically are equal, the perpendicu- 
lars to the sides of the triangle at these points are concurrent. (T. G. de Oppel, 
“Analysis Triangulorum,” p. 32, 1746.) 
23. If on the sides of a triangle ABC, equilateral triangles LBC, MCA, 
NAB be described externally, AL, BM, CN are equal and concurrent. 
24. If on the sides of a triangle ABC, equilateral triangles I/BC, M’CA, 
N’AB be described internally, AL’, BM’ CN’ are equal and concurrent. (Dr. 
J.S. Mackay gives 24 in Vol. XV of Proceedings of Edinburgh Mathematical 
Society and attributes 23 to T. 8. Davies in Gentleman’s Diary for 1830, p. 36.) 
25. If A” B’C” be Nagel’s triangle, then perpendiculars from A, B, and C 
upon B’C”, C’ A’, A’ B”, respectively, are concurrent. 
26. Perpendiculars from A’’, B’, C’ upon BC, CA, AB, respectively, are 
concurrent. 
27. If A’, B’, C’ be the midpoints of the arcs subtended by BC, CA, AB, 
respectively, then perpendiculars from A’, B’, C’ upon B’C”, C’A’, A” B”, 
respectively, are concurrent. 
28. Perpendiculars from A’, B’’, C’ upon B’C’, CA’, A’B’, respectively, 
are concurrent. 
29. If distances equal to 2r (diameter of the inscribed circle) be laid off 
from the vertices on each of the altitudes, three points Ai’, Biv, Civ are obtained. 
Perpendiculars from A, B, C upon Biv Civ, Civ Aiv, Aiv Biv, respectively, are con- 
current. 
30. Perpendiculars from Ai’, Biv, Civ upon BC, CA, AB, respectively, are 
concurrent. (Nos. 25, 27 and 29 are given in Schwatt’s ‘‘Geometric Treatment 
of Curves,” pages 40, 43 and 44. Nos. 26, 28 and 30 are direct consequences of 
the orthologous relation of the triangles. See No. 20.) 
31. The perpendiculars from the middle points of the sides of Brocard’s first 
triangle upon the corresponding sides of the triangle A BC are concurrent. 
« 32, The lines joining the middle points of the sides of a triangle with those 
of the segments towards the angles of the corresponding altitudes meet in a point 
and bisect each other. 
