Geometrical Propositions Demonstrated, Or, a Supplement to Euclid: Being a Key to the Exercises Appended to Euclid's Elements

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Whittaker and Company, 1840 - 94 sider

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Side 29 - The perpendicular is the shortest straight line which can be drawn from a given point to a given straight line ; and of others that which is nearer to the perpendicular is less than the more remote, and the converse; and not more than two equal straight lines can be drawn from the given point to the given straight line, one on each side of the perpendicular.
Side 48 - Prove that in any quadrilateral the sum of the squares of the sides is equal to the sum of the squares of the diagonals, increased by four times the square of the line joining the middle points of the diagonals.
Side 66 - If from any point in the diameter a perpendicular be drawn meeting any chord, or chord produced, the rectangle contained by the segments of the diameter, is equal to the sum or difference of the square of the perpendicular, and the rectangle contained by the segments of the chord, according as the perpendicular meets the chord within or without the circle.
Side 41 - Prove analytically that in any right triangle the straight line drawn from the vertex of the right angle to the middle point of the hypotenuse is equal to half the hypotenuse.
Side 85 - ... and if from these points two straight lines be drawn to any point whatsoever in the circumference of the circle, the ratio of these lines will be the same with the ratio of the segments intercepted between, the two first-mentioned points and the circumference of the circle.
Side 19 - In a right triangle the perpendicular drawn from the vertex of the right angle to the hypothenuse divides the triangle into two triangles similar to the whole triangle and to each other.
Side 41 - In any right triangle, the straight line drawn from the vertex of the right angle to the middle of the hypotenuse is equal to one-half the hypotenuse (I.
Side 80 - If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.
Side 87 - For the rectangle under the sum and difference of the two sides is equal to the rectangle under the sum and difference of the two segments. Therefore, forming the sides of these rectangles into a proportion, their sums and differences will be found proportional.
Side 68 - AB describe a segment of a circle containing an angle equal to the given angle, (in.

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