One of the proofs done by Euclid in his “Elements” book, where he laid the foundation for geometry, is that if you draw lines parallel to the sides of a parallelogram and through any point on the diagonal of a parallelogram, then the two parallelograms above and below that diagonal are equal in area. We’ll repeat the proof here.

## Probelm

*ABCD* is a parallelogram, and *DB* is one of its diagonals. *FH*||*AD* and *EG*||*DC*. Show that the area of *AEKF* is equal to the area of *HKGC*.

## Strategy

We first note that because *EG* and *FH* were drawn parallel to two pairs of parallel sides (AB and DC, AD and BC, respectively), all four quadrilaterals *AEKF*, *EDHK, FKGB *and *HKGC* are themselves parallelograms.

To find the area of a parallelogram, we usually need to find the length of a side, and the height – but none of these are available here. So we will need to use another strategy. Since we don’t actually need to find the area of *AEKF* or *HKGC*, just to show that they are equal, we will try to do this using another technique we’ve used in the past, here or here – subtracting known areas to arrive at the unknown ones.

We have previously shown, in the process of proving that the two pairs of opposite angles of a parallelogram are congruent, that the parallelogram’s diagonal bisects it into two congruent triangles, so the area of Δ*ABD* is the same as the area of Δ*BDC*.

But as the smaller quadrilaterals, *EDHK* and *FKGB* themselves parallelograms that also share the same diagonal, by the above logic the area of Δ*EDK* is the same as the area of Δ*KDH*. And the area of Δ*BFK* is the same as the area of Δ*BGK*.

Subtracting the equal areas of the smaller triangles from the equal areas of the larger triangle leaves us with the areas of the two parallelograms which are equal.

## Proof

(1) AD||BC //Given. ABCD is a parallelogram

(2) AB||DC //Given. ABCD is a parallelogram

(3) *FH*||*AD* // Given

(4) *EG*||*DC* // Given

(5) *EDHK* is a parallelogram //(3), (4), defintion of a parallelogram

(6) *FKGB* is a parallelogram //(3), (4), defintion of a parallelogram

(7) Δ*EDK* ≅ Δ*KDH* //(5), The diagonal of a parallelogram bisects it into two congruent triangles

(8) Δ*BFK* ≅ Δ*BGK* //(6), The diagonal of a parallelogram bisects it into two congruent triangles

(9) Δ*ABD* ≅ Δ*BDC* //(The diagonal of a parallelogram bisects it into two congruent triangles

(10) Area(ABD) = Area(BDC) //areas of congruent triangles equal

(11) Area(BFK)=Area(BGK) //areas of congruent triangles equal

(12) Area(EDK)=Area(KDH) //areas of congruent triangles equal

(13) Area(ABD)-Area(BFK)-Area(EDK)=Area(BDC)-Area(BGK)-Area(KDH) //(10), (11), (12), Transitive and Subtraction properties of equality

(14) Area(S_{1})=Area(S_{2})