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1. Similar rectilineal figures, are those which are equiangular, and have the sides about the equal angles proportional.
2. The homologous, or like sides, of similar figures, are those which are opposite to equal angles.
3. Two figures are said to have their fides reciprocally proportional, when the first confequent, and second antecedent, of the four terms, are both fides of the fame figure.
4. Of three proportional quantities, the middle one is faid to be a mean proportional between the other two; and the last a third proportional to the first and second.
5. Of four proportional quantities, the last is faid to be a fourth proportional to the other three, taken in order.
6. If any number of magnitudes be continually proportional, the ratio of the first and third is said to be duplicate that of the first and second; and the ratio of the first and fourth, triplicate that of the first and second.
7. And of any number of magnitudes, of the same kind, taken in order, the ratio of the first to the last, is said to be compounded of the ratio of the first to the second, of the second to the third, and so on, to the last,
8. A right line is said to be divided in extreme and mean ratio, when the whole line is to the greater segment, as the greater segment is to the less.
PROP. PROP. 1.
Triangles and parallelograms, having the fame altitude, are to each other as their bases.
Let the triangles ABC, ACD, and the parallelograms EC, CF have the same altitude, or be between the same parallels BD, EF; then will the base bc be to the base CD, as the triangle ABC is to the triangle ACD, or as the parallelogram ec is to the parallelogram cf.
For, in bd produced, take any number of parts whatever BG, GH, each equal to BC; and DK, KL, any number whatever, each equal to CD; and join AG, AH, AK and AL:
Then, because CB, BG, GH are all equal to each other, the triangles AHG, AGB, ABC will also be equal to each other (II. 5.); and whatever' multiple the base hc is of the base BC, the same multiple will the triangle Ahc be of the triangle ABC.
And, for the same reason, whatever multiple the base ic is of the base cd, the same multiple will the triangle ALC be of the triangle ADC.
If, therefore, the base hc be equal to the base cl, the triangle Ahc will be equal to the triangle Alc; and if greater, greater; and if less, less.
But the base hc, and the triangle anc, are any equimultiples whatever of the base bc, and the triangle ABC; and the base ci and the triangle Alc are any equimultiples, whatever of the base cd and the triangle ADC; whence the base Bc is to the base cd, as the triangle ABC is to the triangle ACD (V. Def. 5.)
Again, because the parallelogram ce is double the triangle ABC (I. 32.), and the parallelogram cr is double the triangle Adc, the triangle ABC will be to the triangle ADC as the parallelogram ce is to the parallelogram CF (V. 13.)
But, it has been shewn, that the base bc is to the base CD, as the triangle ABC is to the triangle ADC; therefore the base bc is also to the base cd, as the parallelogram ce is to the parallelogram cf.
Q. E. D. COROLL. 'Triangles and parallelograms, having equal altitudes, are to each other as their bases.
PRO P. II.
Triangles and parallelograms, having equal bases, are to each other as their altitudes.
Let ABC, der be two triangles, having the equal bases AB, DE, and whose altitudes are CH, FI; then will the
triangle triangle ABC have the same ratio to the triangle DEF, as CH has to FI.
For make BP perpendicular to AB, and equal to ch (I. 11 and 3.); and in BP take BQ equal to FI, and join AP, AQ and CP.
Then, because Bp is equal to ch, and the base AB is common, the triangle ABP will be equal to the triangle ABC (II. 3.)
And, because AB is equal to DE, and be to FI, the triangle ABQ will also be equal to the triangle DEF (II. 5.)
But the triangle ABP is to the triangle ABQ as Bp is to BQ (VI. 1.); therefore the triangle ABC is also to the triangle der as be is to BQ, or as ch to Fi (V.9.)
And, since parallelograms, having the same bases and altitudes, are the doubles of these triangles, they will, likewise, have to each other the same ratio as their altitudes.
R. E. D. COR. 1. If the bases of equal triangles are equal, the altitudes will also be equal; and if the altitudes are equal, the bases will be equal.
Cor. 2. From this, and the former proposition, it also appears, that rectangles which have one fide in each equal, are proportional to their other fides.
If a right line be drawn parallel to one of the sides of a triangle, it will cut the other sides proportionally : and if the sides be cut proportionally, the line will be parallel to the remaining side of the triangle.
Let ABC be a triangle, and De be drawn parallel to the fide BC; then will ad be to DB, as Ae is to EC.
For join the points B, E, and c, D:
Then, because the triangles DBE, Dce are upon the fame base de, and between the fame parallels DE, BC, they will be equal to each other (I. 31.)
And, fince equal magnitudes have the same ratio to the fame magnitude (V.9.), the triangle DBE will be to the triangle DAE, as the triangle Dce is to the triangle DAE.
But triangles of the fame altitude are to each other as their bases (VI. 1.); whence the triangle DBE will be to the triangle DAE as DB is to da.
For the same reason, the triangle DCE will be to the triangle DAE, as ec is to fa.