Brahmagupta's Theorem

Eric W. Weisstein
January 4, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/BrahmaguptasTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BrahmaguptasTheorem.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<PlaneGeometry`;

Theorem

In a \href{cyclic quadrilateral} $ABCD$ having perpendicular diagonals AC⊥BD, the perpendiculars to the sides through point $T$ of intersection of the diagonals (the\href{anticenter}) always bisects the opposite side (so $M_{AB}$,$M_{BC}$,$M_{CD}$,and $M_{DA}$ are the\hrefacc{midpoints}{Midpoint} of the corresponding sides of the\href{quadrilateral}).

Figure

ang={.2,1.4,3.4};
q=CyclicQuadrilateralWithPerpendicularDiagonals[ang];
v=Vertices[q];
m=Maltitudes[q];
mid=Midpoints[q];
redpts={
mid[[1]],(Subscript[M, AB]),mid[[2]],(Subscript[M, BC]),mid[[3]],(Subscript[M, CD]),mid[[4]],(Subscript[M, AD])
};
pts={v[[1]],A,v[[2]],B,v[[3]],C,v[[4]],D};
bluepts={Anticenter,T};
Show[Graphics[{
{
Red,
Sides[q],
EqualSideTicksAboutMidpoints[q,.05],
PointLabels[q,redpts,.03]
},
PointLabels[q,pts,.03],
Circle[{0,0},1],
{
Blue,
m,
PerpSquare[#,.08,3]&/@Transpose[{m,Sides[q]}],
PointLabels[q,bluepts,.03]
}
}],AspectRatio->Automatic,PlotRange->All]