Similar Polygons Composed of Similar Triangles

Theorem

As Euclid defined it:

Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to corresponding side.

(The Elements: Book VI: Proposition $20$)

Proof

Let $ABCDE$ and $FGHKL$ be similar polygons, and let $AB$ correspond to $FG$.

We need to show that $ABCDE$ and $FGHKL$ are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes.

Also that the area of the polygon $ABCDE$ has to the polygon $FGHKL$ a ratio duplicate of $AB : FG$.

Join up $BE, EC, GL, LH$.

Since $ABCDE$ and $FGHKL$ are similar, $\angle BAE = \angle GFL$.

From Book VI Definition 1: Similar Rectilineal Figures, $BA : AE = GF : FL$.

Thus from Triangles with One Equal Angle and Two Sides Proportional are Similar, $\triangle ABE$ is similar to $\triangle FGL$.

So $\angle ABE = \angle FGL$.

But $\angle ABC = \angle FGH$ because $ABCDE$ and $FGHKL$ are similar.

So $\angle EBC = \angle LGH$.

Because $\triangle ABE$ is similar to $\triangle FGL$, $EB : BA = LG : GF$

Also, because $ABCDE$ and $FGHKL$ are similar, $AB : BC = FG : GH$.

So from Equality of Ratios Ex Aequali, $EB : BC = LG : GH$.

So from Triangles with One Equal Angle and Two Sides Proportional are Similar, $\triangle EBC$ is similar to $\triangle LGH$.

For the same reason, $\triangle ECD$ is similar to $\triangle LHK$.

So $ABCDE$ and $FGHKL$ have been divided into similar triangles, and into triangles equal in multitude.

Now let $AC, FH$ be joined.

Because $ABCDE$ and $FGHKL$ are similar, $\angle ABC = \angle FGH$.

From Triangles with One Equal Angle and Two Sides Proportional are Similar $\triangle ABC$ is similar to $\triangle FGH$.

Therefore $\angle BAC = \angle GFH$ and $\angle BCA = \angle GHF$.

Also, we have that $\angle BAM = \angle GFN$, and $\angle ABM = \angle FGN$.

So from Sum of Angles of Triangle Equals Two Right Angles $\angle AMB = \angle FGN$ and so $\triangle ABM$ is similar to $\triangle FGN$.

Similarly we can show that $\triangle BMC$ is similar to $\triangle GNH$.

Therefore $AM : MB = FN : NG$ and $BM : MC = FN : NH$.

But from Areas of Triangles and Parallelograms Proportional to Base, $AM : MC = \triangle ABM : \triangle MBC$.

So from Sum of Components of Equal Ratios $\triangle ABM : \triangle MBC = \triangle ABE : \triangle CBE$.

But $\triangle ABM : \triangle MBC = AM : MC$.

So $\triangle ABE : \triangle CBE = AM : MC$.

For the same reason, $FN : NH = \triangle FGL : \triangle GLH$.

As $AM : MC = FN : NH$, it follows that $\triangle ABE : \triangle BEC = \triangle FGL : \triangle GLH$.

Similarly $\triangle ABE : \triangle FGL = \triangle BEC : \triangle GLH$.

We now join $BD$ and $GK$, and by a similar construction show that $\triangle BEC : \triangle LGH = \triangle ECD : \triangle LHK$.

From Sum of Components of Equal Ratios it follows that $ABE : FGL = ABCDE : FGHKL$.

But from Ratio of Areas of Similar Triangles $\triangle ABE$ has to $\triangle FGL$ a ratio duplicate of $AB : FG$.

Therefore the area of the polygon $ABCDE$ has to the polygon $FGHKL$ a ratio duplicate of $AB : FG$.

$\blacksquare$

Historical Note

This is Proposition 20 of Book VI of Euclid's The Elements.