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| Triangle Area (Posted on 2008-11-28) |
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Let ABC be any triangle. Let D be a point on side AB and let E be a point on side AC. Draw lines CD and BE and call their intersection F. Triangle ABC is then divided into three smaller triangles BDF, CEF, BCF and a quadrilateral ADFE.
Let the area of BDF equal r, the area of CEF equal q, and the area of BCF equal p. Express the area of the whole triangle ABC in terms of p, q, and r.
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Submitted by Brian Smith
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Solution:
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Draw line AF. Let the area of AEF equal x and the area of ADF equal y. Using the Area Principle, equations in x,y,p,q,r can be made:
AD/DB = x/q = (y+r)/p
AE/EC = y/r = (x+q)/p
Simplifying a little yields:
px = q*y + q*r
py = r*x + q*r
Solving this system yields:
x = (p*q*r + q^2 * r)/(p^2 - r*q)
y = (p*q*r + q * r^2)/(p^2 - r*q)
The area of triagle ABC equals x+y+p+q+r. Substituting the expressions for x and y yields:
ABC = p*(p + q)*(p + r)/(p^2 - r*q)
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