Brahmagupta Theorem

Theorem

If a cyclic quadrilateral has diagonals which are perpendicular, then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.

Specifically:

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular, crossing at $M$.

Let $EF$ be a line passing through $M$ and crossing opposite sides $BC$ and $AD$ of $ABCD$.

Then $EF$ is perpendicular to $BC$ if and only if $F$ is the midpoint of $AD$.

Proof Sufficient Condition

Suppose that $EF$ is perpendicular to $BC$.

We need to prove that $AF = FD$.

Thus:

 $\ds \angle FAM$ $=$ $\ds \angle CAD$ producing $AF$ to $D$ and $AM$ to $C$ $\ds$ $=$ $\ds \angle CBD$ Angles in Same Segment of Circle are Equal: both subtend $CD$ $\ds$ $=$ $\ds \angle CBM$ producing $BM$ to $D$ $\ds$ $=$ $\ds \angle CME$ both are complements of $\angle BCM$ $\ds$ $=$ $\ds \angle FMA$ Vertical Angle Theorem

Then by Triangle with Two Equal Angles is Isosceles, it follows that $AF = FM$.

Similarly:

 $\ds \angle FDM$ $=$ $\ds \angle ADB$ producing $DF$ to $A$ and $DM$ to $B$ $\ds$ $=$ $\ds \angle ACB$ Angles in Same Segment of Circle are Equal: both subtend $AB$ $\ds$ $=$ $\ds \angle BCM$ producing $BM$ to $A$ $\ds$ $=$ $\ds \angle BME$ both are complements to $\angle CBM$ $\ds$ $=$ $\ds \angle DMF$ Vertical Angle Theorem

Then by Triangle with Two Equal Angles is Isosceles, it follows that $FD = FM$.

So $AF = FD$, as we needed to show.

$\Box$

Necessary Condition

Now suppose that $AF = FD$.

We now need to show that $EF$ is perpendicular to $BC$.

From Thales' Theorem (indirectly) we have that $AF = FM = FD$.

So:

 $\ds \angle EBM$ $=$ $\ds \angle CBD$ producing $EB$ to $C$ and $BM$ to $D$ $\ds$ $=$ $\ds \angle CAD$ Angles in Same Segment of Circle are Equal: both subtend $CD$ $\ds$ $=$ $\ds \angle FAM$ producing $AM$ to $C$ and $FA$ to $D$ $\ds$ $=$ $\ds \angle AMF$ Isosceles Triangle has Two Equal Angles, and $AF = FM$ $\ds$ $=$ $\ds \angle EMC$ Vertical Angle Theorem

We note the result Sum of Angles of Triangle equals Two Right Angles.

We have that $\angle EBM$ and $\angle ECM$ are complementary, as both are angles in $\triangle CBM$, which is a right triangle.

So $\angle EMC$ and $\angle ECM$ are complementary, which means that $\angle CEM$ must be a right angle.

Hence by definition $EF$ is perpendicular to $BC$, as we were to show.

$\blacksquare$

Source of Name

This entry was named for Brahmagupta.