2019 AIME I Problems/Problem 3
In , , , and . Points and lie on , points and lie on , and points and lie on , with . Find the area of hexagon .
We know the area of the hexagon to be . Since , we know that is a right triangle. Thus the area of is . Another way to compute the area is Then the area of . Preceding in a similar fashion for , the area of is . Since , the area of . Thus our desired answer is
Let be the origin. Noticing that the triangle is a 3-4-5 right triangle, we can see that , and . Using the shoelace theorem, the area is . Shoelace theorem:Suppose the polygon has vertices , , ... , , listed in clockwise order. Then the area of is
You can also go counterclockwise order, as long as you find the absolute value of the answer.
Solution 3 (Easiest, uses only basic geometry too)
Note that has area and is a 3-4-5 right triangle. Then, by similar triangles, the altitude from to has length 3 and the altitude from to has length 4, so , meaning that . -Stormersyle
Knowing that has area 150 and is a 3-4-5 triangle, we can find the area of the smaller triangles , , and and subtract them from to obtain our answer. First off, we know has area since it is a right triangle. To the find the areas of and , we can use Law of Cosines () to find the lengths of and , respectively. Computing gives and . Now, using Heron's Formula, we find and . Adding these and subtracting from , we get -Starsher
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