In the figure, in a circle with diameter AB = 3, CD is a chord that intersects AB at E, with AC = AE = 2. A line EF is drawn through E perpendicular to AC at F, extending FE to intersect BD at G. Please find the ratio EF : EG = ________.
[Analysis and Solution]
The key is to establish a connection with BC, as shown in the diagram:
Clearly, FE // BC, so
EF / BC = AE / AB = 2 / 3.
To find the target ratio EF : EG, we need to determine EG / BC. Given that EG / BC = DE / CD, the essential task is to find DE : CE. The lengths of DE and CE can be determined as follows:
In right triangle ACB, we have
cosA = 2 / 3.
Therefore, applying the cosine theorem in triangle ACE, we have:
CE² = AC² + AE² - 2cosA × AE × AC = 8 / 3
Thus,
CE = 2√6 / 3.
Since AB and CD are intersecting chords, we apply the intersecting chords theorem:
DE × CE = AE × BE = 2 × 1 = 2,
which gives us
DE = √6 / 2.
Consequently, the length of CD is:
CD = CE + DE = 2√6 / 3 + √6 / 2 = 7√6 / 6.
Now, we can compute EG / BC:
EG / BC = DE / CD = (√6 / 2) / (7√6 / 6) = 3 / 7.
Thus, we arrive at the ratio:
EF : EG = (EF / BC) : (EG / BC) = (2 / 3) : (3 / 7) = 14 : 9.
