MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Competitions 141
Problem 47 (Turkish Mathematical Olympiad 2000). Show that it is
possible to cut any triangular prism of infinite length with a plane such that
the resulting intersection is an equilateral triangle. [9, p. 147]
Figure 5.17: Hungarian National Olympiad 1987
Problem 48 (Hungarian National Olympiad 1987). Cut the equilateral
triangle AXY from rectangle ABCD in such a way that the vertex X is on
side BC and the vertex Y in on side CD (Figure 5.17). Prove that among the
three remaining right triangles there are two, the sum of whose areas equals
the area of the third. [306, p. 5]
Figure 5.18: Austrian-Polish Mathematics Competition 1993
Problem 49 (Austrian-Polish Mathematics Competition 1993). Let
∆ABC be equilateral. On side AB produced, we choose a point P such that
A lies between P and B. We now denote a as the length of sides of ∆ABC;
r1 as the radius of incircle of ∆PAC; and r2 as the exradius of ∆PBC with
respect to side BC (Figure 5.18). Show that r1 + r2 = a√3. [306, p. 7]
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