MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

132 Mathematical Competitions
Figure 5.5: AMC 1981
Problem 10 (AMC 1981). In Figure 5.5, equilateral ∆ABC is inscribed
in a circle. A second circle is tangent internally to the circumcircle at T and
tangent to sides AB and AC at points P and Q, respectively. Show that the
ratio of the length of PQ to the length of BC is 2 : 3. [11, p. 56]
Problem 11 (AMC 1983). Segment AB is a diameter of a unit circle and
a side of an equilateral triangle ABC. The circle also intersects AC and BC
at points D and E, respectively. Show that the length of AE is equal to √ 3.
[22, p. 2]
Figure 5.6: AMC 1988
Problem 12 (AMC 1988). In Figure 5.6, ABC and A ′ B ′ C ′ are equilateral
triangles with parallel sides and the same center. The distance between side
BC and B ′ C ′ is 1 the altitude of ∆ABC. Show that the ratio of the area of
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∆A ′ B ′ C ′ to the area of ∆ABC is 1 : 4. [22, p. 36]