# Area of Equilateral Triangle | AIME I, 2015 | Question 4

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2015 from Geometry, based on Area of Equilateral Triangle (Question 4).

## Area of Triangle - AIME I, 2015

Point B lies on line segment AC with AB =16 and BC =4. Points D and E lie on the same side of line AC forming equilateral triangle ABD and traingle BCE. Let M be the midpoint of AE, and N be the midpoint of CD. The area of triangle BMN is x. Find $x^{2}$.

• is 107
• is 507
• is 840
• cannot be determined from the given information

### Key Concepts

Algebra

Theory of Equations

Geometry

AIME, 2015, Question 4

Geometry Revisited by Coxeter

## Try with Hints

First hint

Let A(0,0), B(16,0),C(20,0). let D and E be in first quadrant. then D =$(8,8\sqrt3)$, E=$(18,2\sqrt3$).

Second Hint

M=$(9,\sqrt3)$, N=($14,4\sqrt3$), where M and N are midpoints

Final Step

since BM, BN, MN are all distance, BM=BN=MN=$2\sqrt13$. Then, by area of equilateral triangle, x=$13\sqrt3$ then$x^{2}$=507.

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