PROP. VIII. THEOR.

In a right angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.*

Let ABC be a right angled triangle, having the right angle BAC; and from the point A let AD be drawn perpendicular to the base BC: the triangles ABD, ADC are similar to the whole triangle ABC, and to one another.

A

Because the angle BAC is equal to the angle ADB, each of them being a right angle, and that the angle at B is common to the two triangles ABC, ABD; the remaining angle ACB is equal to the remaining angle BAD (32. 1.): therefore the triangle ABC is equiangular to the triangle ABD, and the sides about their equal angles are proportionals (4. 6.); wherefore B the triangles are similar (1. Def. 6.):

D C

in the like manner it may be demonstrated, that the triangle ADC is equiangular and similar to the triangle ABC and the triangles ABD, ADC, being both equiangular and similar to ABC, are equiangular and similar to each other. Therefore, in a right angled, &c. Q. E. D.

COR. From this it is manifest, that the perpendicular drawn from the right angle of a right angled triangle to the base, is a mean proportional between the segments of the base and also that each of the sides is a mean proportional between the base, and its segment adjacent to that side: because in the triangles BDA, ADC, BD is to DA as DA to DC (4. 6.); and in the triangles ABC, DBA, BC is to BA, as BA to BD (4. 6.); and in the triangles ABC, ACD, BC is to CA as CA to CD (4. 6.).

* See Note.

PROP. IX. PROB.

FROM a given straight line to cut off any part required.*

Let AB be the given straight line; it is required to cut off any part from it.

From the point A draw a straight line AC making any angle with AB; and in AC take any point D, and take AC the same multiple of AD, that AB is of the part which

is to be cut off from it: join BC, and draw DE parallel to it: then AE is the part required to be cut off.

A

Because ED is parallel to one of the sides of E the triangle ABC, viz. to BC, as CD is to DA, so is (2. 6.) BE to EA; and, by composition (18. 5.) CA is to AD as BA to AE: but CA is a multiple of AD; therefore (D. 5.) BA is the same multiple of AE: whatever part therefore AD is of AC, AE is the same part of AB: B wherefore, from the straight line AB the part required is cut off. Which was to be done..

PROP. X. PROB.

C

To divide a given straight line similarly to a given divided straight line, that is, into parts that shall have the same ratios to one another which the parts of the divided given straight line have.

Let AB be the straight line given to be divided, and AC the divided line; it is required to divide AB similarly to AC.

Let AC be divided in the points D, E; and let AB, AC be placed so as to contain any angle, and join BC, and through the points D, E draw (31. 1.) DF, EG parallels to it; and through D draw DHK parallel to AB: therefore each of the figures FH, HB, is a parallelogram; wherefore DH is equal (34. 1.) to FG, and

* See Note.

F

A

H

E

HK to GB: and because HE is parallel to KC, one of the sides of the triangle DKC, as CE to ED, so is (2. 6.) KH to HD: but KH is equal to BG, and HD to GF; therefore as CE to ED, so is BG to GF; again, because FD is parallel to EG, one G of the sides of the triangle AGE, as ED to DA, so is GF to FA; but it has been proved that CE is to ED as BG to GF; and as ED to DA, so GF to FA: therefore the given straight line AB is divided similarly to AC. Which was to be done.

B

K

C

PROP. XI. PROB.

To find a third proportional to two given straight lines.

Let AB, AC be the two given straight lines, and let them be placed so as to contain any angle; it is required to find a third proportional to AB, AC.

Produce AB, AC to the points D, E and make BD equal to AC; and having joined BC, through D draw DE parallel to it (31. 1.).

B

A

Because BC is parallel to DE, a side of the triangle ADE, AB is (2. 6.) to BD, as AC to CE: but BD is equal to AC; as therefore AB to AC, so is AC to CE. Wherefore, to the two given straight lines AB, AC a third pro- D portional CE is found. Which was to be done.

C

E

PROP. XII. PROB.

To find a fourth proportional to three given straight lines.

Let A, B, C be the three given straight lines; it is required to find a fourth proportional to A, B, C.

Take two straight lines DE, DF, containing any angle EDF:

and upon these make DG equal
to A, GE equal to B, and DH
equal to C; and having joined GH,
draw EF parallel (31. 1.) to it
through the point E: and because
GH is parallel to EF, one of the
sides of the triangle DEF, DG is G
to GE, as DH to HF (2. 6.); but
DG is equal to A, GE to B, and
DH to C; therefore, as A is to B,
so is C to HF: wherefore to the

E

D

A

B

C

F

three given straight lines A, B, C, a fourth proportional HF is found. Which was to be done.

PROP. XIII. PROB.

To find a mean proportional between two given. straight lines.

Let AB, BC be the two given straight lines; it is required to find a mean proportional between them.

Place AB, BC in a straight line, and upon AC describe the semicircle ADC, and from the point

B draw (11. 1.) BD at right angles to AC, and join AD, DC.

Because the angle ADC in a semicircle is a right angle (31. 3.), and because in the right angled triangle ADC, DB is drawn from the right angle perpendicular to the base, ĎB A

D

B

C

is a mean proportional between AB, BC, the segments of the base (Cor. 8.6.): therefore between the two given straight linesAB, BC a mean proportional DB is found. Which was to be done.

PROP. XIV. THEOR.

EQUAL parallelograms which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional: and parallelograms that have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.

F

Let AB, BC be equal parallelograms, which have the angles at B equal, and let the sides DB, BE be placed in the same straight line wherefore also FB, BG are in one straight line (14. 1.): the sides of the parallelograms AB, BC, about the equal angles, are reciprocally proportional; that is, DB is to BE, as GB to BF. Complete the parallelogram FE: and because the parallelogram AB is equal to BC, and that FE is A another parallelogram, AB is to FE, as BC to FE (7. 5.): but as AB to FE, so is the base DB to BE (1. 6.); and as BC to FE, so is the base GB to BF: therefore as DB to BE, so is GB to BF (11. 5.). Wherefore the sides of the parallelograms AB, BC

E

D

B

G

C

about their equal angles are reciprocally proportional. But, let the sides about the equal angles be reciprocally proportional, viz. as DB to BE, so GB to BF; the parallelogram AB is equal to the parallelogram BC.

Because as DB to BE, so is GB to BF; and as DB to BE, so is the parallelogram AB to the parallelogram FE; and as GB to BF, so is the parallelogram BC to the parallelogram FE; therefore as AB to FE, so BC to FE (9. 5.): wherefore the parallelogram AB is equal (9. 5.) to the parallelogram BC. Therefore, equal parallelograms, &c. Q. E. D.