PROPOSITION XVII. THEOREM.

If three straight lines be proportionals, the rectangle contained by the extremes is equal to the square on the mean.

Let the three st. lines A, B, C be proportionals, and let A be to B as B is to C.

Then must rect. A, C=sq. on B.

Take D=B.

Then A is to B as B is to C,

.. A is to B as D is to C,

and.. rect. A, C=rect. B, D,

that is, rect. A, C=sq. on B.

And Conversely,

V. 6.

VI. 16.

If the rectangle contained by the extremes be equal to the square on the mean, the three straight lines are proportionals.

Let A, B, C be three straight lines such that

rect. A, C=sq. on B.

Then must A be to B as B is to C.

For, the same construction being made,

rect. A, C=sq. on B,

and B=D,

.. rect. A, C=rect. B, D ; and.. A is to B as D is to C, that is, A is to B as B is to C.

VI. 16.

V. 6. Q. E. D.

Upon a given straight line to describe a rectilinear figure similar and similarly situated to a given rectilinear figure.

H

E

Let AB be the given st. line, and CDEF the given rectil. fig. of four sides.

It is required to describe on AB a fig. similar and similarly situated to CDEF.

Join DF, and at A and B, make ▲ BAG

L ABG= L CDF;

Δ

= 4 DCF, and

then ▲ BAG is equiangular to ▲ DCF. At G and B, make ▲ BGH ▲ DFE, and 2 GBH= ▲ FDE ; then ▲ GHB is equiangular to ▲ FED.

Then AGB= ▲ CFD, and ▲ BGH= 2 DFE,

Also,

.. rectil. fig. ABHG is equiangular to fig. CDEF.

BAG is equiangular to ▲ DCF,

.. BA is to AG as DC is to CF;

VI. 4.

and ·.· ▲ BGH is equiangular to ▲ DFE,

.. GB is to GH as FD is to FE.

VI. 4.

Also, AG is to GB as CF is to FD.

.. AG is to GH as CF is to FE. Similarly, it may shown that

GH is to HB as FE is to ED,

and that HB is to BA as ED is to DC.

.. the rectil. figs. ABHG and CDEF are similar.

V. 21.

NEXT. Let it be required to describe on AB a fig., similar and similarly situated to the rectil. fig. CDKEF.

H

B

Join DE, and on AB describe the fig. ABHG, similar and similarly situated to the quadrilateral CDEF.

At B and H make ▲ HBL= ▲ EDK, and ▲ BHL= ¿ DEK ;

then ▲ HLB is equiangular to ▲ EKD.

Δ

Then the figs. ABHG, CDEF are similar,

.. ▲ GHB=L FED;

and we have made ▲ BHL= ▲ DEK ;

.. whole ▲ GHL-whole ▲ FEK.

For the same reason, ▲ ABL= 4 CDK.

L

Thus the fig. AGHLB is equiangular to fig. CFEKD.

.. GH is to HB as FE is to ED:

Ax. 2.

Again, the figs. AGHB, CFED are similar,

also we know that HB is to HL as ED is to EK, .. GH is to HL as FE is to EK.

VI. 4.

V, 21.

For the same reason, AB is to BL as CD is to DK.

VI. 4.

And BL is to LH as DK is to KE;

.. the five-sided figs. AGHLB, CFEKD are similar.

In the same way a fig. of six or more sides may be described,

on a given line, similar to a given fig.

Q. E. F.

PROPOSITION XIX. THEOREM.

Similar triangles are to one another in the duplicate ratio of their homologous sides.

Let ABC, DEF be similar ▲S,

having 4 s at A, B, C= 4 s at D, E, F respectively,
so that BC and EF are homologous sides.

Then must ▲ ABC have to ▲ DEF the duplicate ratio of that which BC has to EF.

Suppose ▲ DEF to be applied to ▲ ABC, so that

E lies on B, ED on BA, and .. EF on BC.

Let P and Q be the pts. in BA, BC on which D and F fall.

Join AQ.

Then ▲ ABC is to ▲ ABQ as BC is to BQ,

VI. 1.

and ▲ ABQ is to ▲ PBQ as AB is to BP.

VI. 1.

But AB is to BP as BC is to BQ,

VI. 4.

V. 5.

V. 5.

..A ABQ is to ▲ PBQ as BC is to BQ.
Hence ABC is to ▲ ABQ as ▲ ABQ is to ▲ PBQ.
..A ABC has to ▲ PBQ the duplicate ratio

of ▲ ABC to ▲ ABQ;

..▲ ABC has to ▲ PBQ the duplicate ratio

of BC to BQ.

VI. Def. 2.

V. 5.

that is, ABC has to ▲ DEF the duplicate ratio of BC to EF.

Q. E. D.

COR. If MN be a third proportional to BC and EF, BC has to MN the duplicate ratio of BC to EF, and.. BC is to MN as A ABC is to A DEF.

VI. Def. 2.

Exercises on Proposition XIX.

Ex. 1. Prove this Proposition without drawing any line inside either of the triangles.

Ex. 2. In the figure, if BC be equal to FD, shew that the triangles will be in the ratio of AB to EF.

Ex. 3. Cut off the third part of a triangle by a straight line parallel to one of its sides.

Ex. 4. AB, AC are bisected in D and E. Prove that the quadrilateral DBCE is equal to three times the triangle ADE.

Ex. 5. ABC is a line passing through the centre of the circle BCD, and AD a tangent to the circle. If CE be drawn parallel to BD, shew that the triangles ACD, ACE are to one another as AC to AB.

Ex. 6. A straight line drawn parallel to the diagonal BD of a parallelogram ABCD meets AB, BC, CD, DA, in E, F, G, H. Prove that the triangles AFG, CEH are equal.

Ex. 7. If two triangles have an angle equal, and be to each other in the duplicate ratio of adjacent sides, they are similar.

Ex. 8. The circle B'C (centre 0) touches the circle ABC internally, and AB'B touches B'C in B'. Shew that if BD be perpendicular to the common diameter, AB, B' divides AB into segments, which are in the duplicate ratio of OC to OD.

Ex. 9. From the extremities A, B, of the diameter of a circle, perpendiculars AY, BZ, are let fall on the tangent at any point C. Prove that the areas of the triangles ACY, BCZ are together equal to that of the triangles ACB.

Ex. 10. If to the circle, circumscribing the triangle ABC, a tangent at C be drawn, cutting AB produced in D, shew that AD is to DB in the duplicate ratio of AC to CB.

Ex. 11. Construct a triangle which shall be to a given triangle in a given ratio.