PlanetMath: proof of Van Aubel's theorem

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proof of Van Aubel's theorem

(Proof)

$\includegraphics{vanaubel}$

As in the figure, let us denote by the areas of the six component triangles. Given any two triangles of the same height, their areas are in the same proportion as their bases (Euclid VI.1). Therefore

$\displaystyle \frac{y+z}{x}=\frac{u+v}{w} \qquad\frac{w+x}{v}=\frac{y+z}{u} \qquad\frac{u+v}{z}=\frac{w+x}{y}$

and the conclusion we want is

$\displaystyle \frac{y+z+u}{v+w+x}+\frac{z+u+v}{w+x+y}=\frac{y+z}{x}\;.$

Clearing the denominators, the hypotheses are

$\displaystyle w(y+z)$	$\displaystyle =$	$\displaystyle x(u+v)$	(1)
$\displaystyle y(u+v)$	$\displaystyle =$	$\displaystyle z(w+x)$	(2)
$\displaystyle u(w+x)$	$\displaystyle =$	$\displaystyle v(y+z)$	(3)

which imply

$\displaystyle vxz=uwy$

(4)

and the conclusion says that

$\displaystyle x(\underline{wy}+\underline{wz}+uw +\underline{xy}+\underline{xz}+ux+\underline{y^2}+\underline{yz}+uy$

$\displaystyle +vz+uv+v^2+wz+uw+vw+xz+ux+vx)$

equals

$\displaystyle (y+z)(vw+vx+vy+w^2+wx+wy+\underline{wx}+\underline{x^2}+\underline{xy})$

or equivalently (after cancelling the underlined terms)

$\displaystyle x(uw+xz+ux+uy+vz+uv+v^2+wz+uw+vw+ux+vx)$

equals

$\displaystyle (y+z)(vw+vx+vy+w^2+wx+wy)=(y+z)(v+w)(w+x+y)\;.$

i.e.

$\displaystyle x(u+v)(v+w+x)+x(xz+ux+uy+vz+wz+uw)=$

$\displaystyle (y+z)w(v+w+x)+(y+z)(vx+vy+wy)$

i.e. by (1)

$\displaystyle x(xz+ux+uy+vz+wz+uw)=(y+z)(vx+vy+wy)$

i.e. by (3)

$\displaystyle x(xz+uy+vz+wz)=(y+z)(vy+wy)\;.$

Using (4), we are down to

$\displaystyle x^2z+xuy+uwy+xwz=(y+z)y(v+w)$

i.e. by (3)

$\displaystyle x^2z+vy(y+z)+xwz=(y+z)y(v+w)$

i.e.

$\displaystyle xz(x+w)=(y+z)yw\;.$

But in view of (2), this is the same as (4), and the proof is complete.

Remarks: Ceva's theorem is an easy consequence of (4).

"proof of Van Aubel's theorem" is owned by mathcam. [ full author list (2) | owner history (1) ]

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See Also: Van Aubel theorem, proof of Van Aubel theorem

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Cross-references: consequence, Ceva's theorem, imply, denominators, conclusion, Proportion, height, triangles, areas

This is version 5 of proof of Van Aubel's theorem, born on 2003-09-20, modified 2006-10-04.
Object id is 4736, canonical name is ProofOfVanAubelsTheorem.
Accessed 2165 times total.

Classification:

51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)

Pending Errata and Addenda

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