1974 IMO Problems/Problem 2: Difference between revisions
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Since this is an "if and only if" statement, we will prove it in two parts. | Since this is an "if and only if" statement, we will prove it in two parts. | ||
Before we begin, note a few basic but important facts. | |||
1. When two variables <math>x</math> and <math>y</math> are restricted by an equation <math>x+y=k</math> for some constant <math>k</math>, the maximum of their product occurs when <math>x=y=\frac{k}{2}</math>. | |||
2. The triangle inequality states that for a triangle with sides <math>a</math>, <math>b</math>, and <math>c</math> fulfills <math>a + b > c</math>, meaning that <math>a > \frac{c}{2}</math> or <math>b > \frac{c}{2}</math>, which is equivalent to saying that <math>a</math> and <math>b</math> both cannot be less than or equal to <math>\frac{c}{2}</math>. | |||
Part 1: | Part 1: | ||
Revision as of 16:47, 6 January 2019
In the triangle ABC; prove that there is a point D on side AB such that CD
is the geometric mean of AD and DB if and only if
.
Solution
Since this is an "if and only if" statement, we will prove it in two parts.
Before we begin, note a few basic but important facts.
1. When two variables 
and 
are restricted by an equation 
for some constant 
, the maximum of their product occurs when 
.
2. The triangle inequality states that for a triangle with sides 
, 
, and 
fulfills 
, meaning that 
or 
, which is equivalent to saying that and both cannot be less than or equal to 
.
Part 1:
