All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
Triangles and Rectangle (Posted on 2005-08-21) Difficulty: 3 of 5
Given a rectangle ABCD with |AB|=a and |BC|=b. There are two distinct equilateral triangles AEF and AGH with points E and G on line BC and points F and H on line CD. What is the product of the areas of the two triangles in terms of a and b?

  Submitted by Bractals    
Rating: 3.3333 (3 votes)
Solution: (Hide)
Analysis:

Define the points A(0,0), B(a,0), C(a,b), and D(0,b).
Let P(a,y) and Q(x,b). If s is the length of a side
of equilateral triangle APQ, then

  s^2 = a^2 + y^2 = x^2 + b^2 = (a - x)^2 + (b - y)^2

Eliminating y from these equations gives a quartic in x

  (x^2 + b^2)*(x^2 - 4*a*x + 4*a^2 - 3*b^2) = 0

having two real roots,

  x = 2*a - b*sqrt(3)   

    ==>   s^2 = 4*(a^2 + b^2 - a*b*sqrt(3))

    ==>   Area(AEF) = sqrt(3)*(a^2 + b^2 - a*b*sqrt(3))

  and

  x = 2*a + b*sqrt(3)   

    ==>   s^2 = 4*(a^2 + b^2 + a*b*sqrt(3))

    ==>   Area(AGH) = sqrt(3)*(a^2 + b^2 + a*b*sqrt(3))

Therefore,

Area(AEF) * Area(AGH) = 3*[a^4 - (a*b)^2 + b^4]

Construction of the triangles: 

Construct circles of radii b and centers B and C intersecting
at two points. Construct rays from point A through these two
intersections and intersecting line CD at points F and H.
Construct circles of radii a and centers C and D intersecting
at two points. Construct rays from point A through these two
intersections and intersecting line BC at points E and G. 

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some ThoughtsPuzzle Thoughts K Sengupta2023-02-06 10:07:22
re(2): AnswerRichard2005-08-22 18:18:08
re: AnswerCharlie2005-08-22 12:30:22
AnswerRichard2005-08-22 06:50:46
Hints/Tipsre: huh?Bractals2005-08-21 09:13:31
Questionhuh?Josh706792005-08-21 06:28:13
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (10)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information