MathDB

Let

ABC

be an arbitrary scalene triangle. Define

\sum

to be the set of all circles

y

that have the following properties:
(i)

y

meets each side of

ABC

in two (possibly coincident) points;
(ii) if the points of intersection of

y

with the sides of the triangle are labeled by

P, Q, R, S, T , U

, with the points occurring on the sides in orders

\mathcal B(B,P,Q,C), \mathcal B(C, R, S,A), \mathcal B(A, T,U,B)

, then the following relations of parallelism hold:

TS \parallel BC; PU\parallel CA; RQ\parallel AB

. (In the limiting cases, some of the conditions of parallelism will hold vacuously; e.g., if

A

lies on the circle

y

, then

T

S

both coincide with

A

and the relation

TS \parallel BC

holds vacuously.)
(a) Under what circumstances is

\sum

nonempty?
(b) Assuming that Σ is nonempty, show how to construct the locus of centers of the circles in the set

\sum

.
(c) Given that the set

\sum

has just one element, deduce the size of the largest angle of

ABC.

(d) Show how to construct the circles in

\sum

that have, respectively, the largest and the smallest radii.

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