[arcæiBazr]  MATHEMATICAL INSTRUCTION IN FRANCE 141 
PARE VEE 
EXAMINATIONS IN THE CONCOURS FOR 
1900. 
(1) WRITTEN. 
MATHÉMATIQUES ELEMENTAIRES.* 
[Time, 7 hours; 7 a.m.-2 p.m]. 
Given two circles, with centres O and O, radii À and À’; these 
circles are exterior to one another, and the common exterior tangents 
are drawn, the points of contact being A and 4’, B and BJ’, whilst the 
points of contact of the common interior tangents are C and C”, D and 
D, the points A and C being on either side of the line of centres where- 
as the contrary takes place for the points A’ and C” if we have, as is 
supposed, R< RF’. The tangents AA’ and CC” cut in £, the tangents 
BB and DD’ cut in F, and the line LF meets the line OO’ in the point 
G; the tangents AA’ and DD’ cut in /, the tangents BB’ and CC" cut 
in / and the line 77 meets the line OO’ in the point A. Consider the 
lines AC, BD and A’C’, BD’, which cross at the point K. 
1°. In order that the lines dC and B’D' become the coincident 
line 7, in which case the lines BD and A'C” become the same line s, it 
is necessary and sufficient that the orthoptic circles of the two given 
circles are orthogonal, which is equivalent to the metric relation 
OO” =2(R* +R") 
(The orthoptic circle of a circle is the circle which is the locus of points 
from which one sees the given circle under a right angle).—The point 
G is then the middle of the segment OO’—The preceding condition is 
supposed fulfilled in all which follows. 
2°. If R is a point of the line 7, the polars of this point with respect 
to the two circles O and O' cut in a point S' situated on the lines; 
3°. The envelope of the line RS is a conic, which is to be de- 
termined by metrical elements; determine the principal tangents. The 
locus of the orthocentre P of the triangle ORS is a conic, of which 
it is required to find some remarkable points ; same question for the 
triangle O'RS. 
4. Suppose RM, RN and RM’, RN’ the tangents drawn from a 
point 2 of the line 7 to the two circles O and O'; the plane being 
oriented in the sense ABCD, let a, B and y, à be the angles, made with 
an axis 7 by the half-lines of the tangents, situated on the same side 
of the line 7 for each of the circles O and O’ (these angles are found 
again at Oand O'); setting 
IE TR RE 8-y 
— 7 — = 
oe c , 
2 SRE 2 
“ 


a 
= 


* See for solutions to questions in this paper Vouvelles Annales de Mathématiques 
(4) IX, 455-67, 1909. 
