Relative Lengths of Lines Outside Circle

Proof

 * Euclid-III-8-1.png

Let $ABC$ be a circle and let $D$ be a point outside the circle.

Let $M$ be the center of the circle and let $E$, $F$, and $C$ be points on the concave circumference.

Draw straight lines $DM$, $DC$, $DE$, and $DF$, then extend $DM$ to $A$ on the circle.

Place $H$, $L$, $K$, and $G$ on the intersections of the circle and $DC$, $DF$, $DE$, and $DA$ respectively.

Then of the straight lines falling on the concave circumference $AEFC$, the line $DA$ through the center is the greatest, with $DE > DF > DC$, and of the straight lines falling on the convex circumference $HLKG$, the line $DG$ is the least, with $DK < DL < DH$.

The proof is as follows:

Join $ME$, $MF$, $MC$, $MH$, $ML$, and $MK$.

Since $AM = EM$, add $DM$ to each, so $AD = EM + MD$.

But we know that $EM + MD > ED$, so it follows that $AD > ED$. Since this is true for any $E$, $AD$ is the greatest line from $D$ to the concave circumference.

Since $EM = FM$, $DM$ is common, and $\angle EMD > \angle FMD$, it follows that $ED > FD$.

Similarly we can show that $FD > CD$. Thus $DA > DE > DF > DC$.

We know that $MK + KD > MD$, and since $MG = MK$, we have $KD > GD$. Since this is true for any $K$, $GD$ is the least line from $D$ to the convex circumference.

Since $K$ is inside $\triangle MLD$, it follows that $ML + LD > MK + KD$. Then since $MK = ML$, we have $DK < DL$.

Similarly we can show that $DL < DH$. Thus $DG < DK < DL < DH$.



Also, from the point $D$ only two equal straight lines fall on the circle $ABC$, one on each side of the line $DA$.

Construct the angle $\angle DMB = \angle KMD$ on the straight line $MD$ at the point $M$. Join $DB$.

Then since $MK = MB$ and $MD$ is common, from Triangle Side-Angle-Side Equality $DK = DB$.

Another straight line equal to $DK$ will not fall on the circle from $D$.

For if this is possible, let $DN$ be this straight line.

Then $DN = DB$, but from what we proved above either $DN > DB$ (or $DN < DB$ depending on where $N$ falls on the circle), a contradiction.

Therefore the point $N$ cannot exist as described, and no more than two equal straight lines fall on the circle from $D$, one on each side of $DA$.