Find three consecutive even integers such that the sum of the least integer and the middle integer is 3030 more than the greates
1 answer:
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I added a figure so you can guide yourself throughout the proof I'm about to write, so I recommend that you download the picture beforehand and have this window and the picture's window open. Alright, let's get started!
Assuming that
is an equilateral triangle according to the wording of the problem, we have that the angles 
.
We also know that the circle in green is Inscribed in.
The following applies to every inscribed circle inside a triangle:
The center of the inscribed circle of a triangle is the intercept of all three angle bisectors of the triangle.
The above theorem implies that the line
is an angle bisector because it goes through the vertex F and the center of the inscribed circle.
The previous statement implies that the angle
.
Now let's work on the
triangle.
Knowing that
(because 
is tangent to the circle at B and 
is a radius of the circle. If you're lost here, remember that a tangent to a circle is always perpendicular to the radius of the circle.) we can then derive that 
(because the sum of the measures of all angles in a triangle is always equal to 180 degrees).
In the same way, we can prove also that:
.
Knowing the above we notice that
.
We're at the last part of our proof here:
Now notice that
subtends the same arc on the circle that 
.
According to the inscribed angle theorem, an angle
inscribed in a circle is half of the central angle 
that subtends the same arc on the circle.
Therefore
We can prove in a similar fashion that:
Therefore all the angles of the
triangle have a measure of 60 degrees, we conclude then that is equilateral.
Assuming that
We also know that the circle in green is Inscribed in
The following applies to every inscribed circle inside a triangle:
The center of the inscribed circle of a triangle is the intercept of all three angle bisectors of the triangle.
The above theorem implies that the line
The previous statement implies that the angle
Now let's work on the
Knowing that
In the same way, we can prove also that:
Knowing the above we notice that
We're at the last part of our proof here:
Now notice that
According to the inscribed angle theorem, an angle
Therefore
We can prove in a similar fashion that:
Therefore all the angles of the