96 
33. The straight lines which join the midpoint of each side of a triangle to 
the midpoint of the corresponding altitude concur at the symmedian point. (Dr. 
F. Wetzig in Schlémlich’s Zeitschrift, XII, 289.) 
34. If two sides of a triangle are divided proportionally the straight lines 
drawn from the points of section to the opposite vertices, will intersect on the 
median from the third vertex. 
35. Every two perpendiculars to the sides of a triangle at points of contact 
of escribed circles external to the same vertex are concurrent with the perpen- 
dicular. to the opposite side at the point of contact of the inscribed circle. There 
will be three such points of concurrency. 
36. If the three sides of a triangle be reflected with respect to any line, the 
three lines through the vertices parallel to the reflexions of the opposite sides are 
concurrent. : 
37. The vertices of ABC are joined to a point O, and a triangle A’B’C’ is 
constructed haying its sides parallel to 40, BO, CO respectively. Lines through 
A’, B’, C’ parallel to the corresponding sides of the triangle 4 BC are concurrent. 
38. If XYZ be any transversal of the triangle ABC, and if AX, BY, CZ 
form the triangle PQR, then AP, BQ, and CR are concurrent. 
39. If D, £, F’be the feet of the altitudes, then the lines connecting A, B, C 
to the middle points of EF, FD, DE, respectively, concur at the symmedian 
point. 
40. The perpendiculars from A, B, C upon EF, FD, DE are concurrent. 
41. Through the vertices of the triangle ABC lines parallel to the opposite 
sides are drawn, meeting the circumcircle in A’, B’, C’. B’C’, C/A’, A’ B’ meet 
BC, CA, AB in P, Q, BR, respectively. AP, BQ, CR are concurrent. 
42. With the same notation as 41, A’P, B’Q, C’R are concurrent. (41 and 
42 occur in St. John’s College Questions, 1890.) 
43. Three circles are drawn each touching two sides of the triangle ABC 
and the cireumcircle internally. The points of contact with the cireumcircle are 
L, M, N, respectively. AL, BM, CN are concurrent. 
44. If in 43 the circles touch the circumcircle externally in L’, M’, N’, 
then L’/A, M’B, N’C are concurrent. (48 and 44 are given by Professor 
de Longchamps, Ed. Times, July, 1890.) 
45.. If a circle touch the sides of the triangle ABC in X, Y, Z, then the 
lines joining the middle points of BC, CA, AB to the middle points of AX, BY, 
CZ, respectively, are concurrent. 
46. If a circle cut the sides of the triangle ABC in X,X’; Y,Y’; ZZ; if 
AX, BY, CZ are concurrent, so also are AX’, BY’, CZ. 
