PROP. VI. THEOR. If two triangles have one angle of the one, equal to one angle of the other, and the sides about the equal angles proportional, the triangles shall be equiangular, and shall have those angles equal which are opposite to the homologous sides, If two triangles have one angle of the one equal to one angle of the other, and the sides about two other angles proportional; then, if the remaining angles be each either less or not less than a right angle, the triangles shall be equiangular, and shall have those angles equal about which the sides are proportional. G B E In a right-angled triangle if a perpendicular be drawn from the right angle to the opposite side, the triangles on each side of it, are similar to the whole triangle and to each other. From a given straight line to cut off any required part. To divide a given undivided straight line into parts proportional to those of a given divided straight line. PROP. XI. PROB. To find a third proportional to two given straight lines. E PROP. XII. PROB. To find a fourth proportional to three given straight lines. PROP. XIV. THEOR. Equal parallelograms which have an angle of the one equal to an angle of the other, have the sides about the equal angles reciprocally proportional and parallelograms which have an angle in the one equal to an angle in the other, and the sides about the equal angles reciprocally proportional, are equal. Equal triangles which have an angle of the one equal to an angle of the other, have the sides about the equal angles reciprocally proportional: and triangles which have an angle of the one equal to an angle of the other, and the sides about the equal angles reciprocally proportional, are equal. E PROP. XVI. THEOR. If four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means: and, if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines are proportional. PROP. XVII. THEOR. If three straight lines be proportional, the rectangle contained by the extremes is equal to the square of the mean: and, if the rectangle contained by the extremes (of three given straight lines) be equal to the square of the mean, the three straight lines are proportional. A C D |