I need help on a geometry challenge from Instagram user gercekboss that really stumps me.
I know some relevant theorems, such as the angle bisector theorem, but I just can't for the life of me figure out how to apply them.
Can anyone give any help? Thank you
Defining $x,y,z$ as shown in the figure above, we can write the following equation from the angle bisector theorem.
$6/x = (5 + y) / (9 + z)$
$5/y = (6 + x)/(9 + z)$
$9 / z = (6 + x)/(5 + y)$
therefore,
$6 (9+z) = x (5 + y)$
$5 (9+z) = y (6 + x)$
$9 (5 + y) = z (6 + x)$
These equation have only one valid solution, namely, $x = 9, y = 5, z = 6$
Hence the sides of the triangle, are $15, 10, 15$
the semi-perimeter is $s =\frac{1}{2} (15 + 10 + 15) = 20 $
Therefore, by Heron's formula, the area is given by
$\text{Area} = \sqrt{ s(s-a)(s-b)(s-c) } =\sqrt{ 20 (5)(5)(10) } = 50 \sqrt{2} $
