382 exercises, solved, upon the 3rd, 4th, 5th, and 6th books of Euclid

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Page 78 - Two points are taken in the diameter of a circle at any equal distances from the center ; through one of these draw any chord, and join its extremities and the other point. The triangle so formed has the sum of the squares of its sides invariable. 156. If chords drawn from any fixed point in the circumference of a circle, be cut by another chord which is parallel to the tangent at that point, the rectangle contained by each chord, and the part of it intercepted between the given point and the given...
Page 19 - If an equilateral triangle be inscribed in a circle, and the adjacent arcs cut off by two of its sides be bisected, the line joining the points of bisection shall be trisected by the sides.
Page 141 - ABD is described, having each of the angles at the base double of the third angle.
Page 116 - The perpendicular from the vertex on the base of an equilateral triangle is equal to the side of an equilateral triangle inscribed in a circle, whose diameter is the base.
Page 9 - C is the centre of a given circle, CA a radius, B a point on a radius at right angles to CA ; join AB and produce it to meet the circle again at D, and let the tangent at D meet CB produced at E: shew that BDE is an isosceles triangle.
Page 52 - Given the vertical angle, the difference of the two sides containing it, and the difference of the segments of the base made by a perpendicular from the vertex ; construct the triangle.
Page 139 - ... any straight line drawn parallel to them will cut the other sides, or those sides produced, proportionally. 351. ABC is a triangle ; it is required to draw from a given point P, in the side AB, or AB produced, a straight line to AC, or AC produced, so that it may be bisected by BC. VI. 3, A. 352. The side BC of a triangle ABC is bisected at D, and the angles ADB, ADC are bisected by the straight lines DE, DF, meeting AB, AC at E, F respectively : shew that EF is parallel to BC. 353. AB is a diameter...
Page 42 - Describe a circle that shall have a given radius and touch a given circle and a given straight line. 184. A circle is drawn to touch a given circle and a given straight line. Shew that the points of contact are always in the same straight line with a fixed point in the circumference of the given circle.
Page 81 - If any chord of a circle be produced equally both ways, and tangents to the circle be drawn on opposite sides of it from its extremities, the line joining the points of contact bisects the given chord.
Page 139 - From a point E in the common base of two triangles ACB, ADB, straight lines are drawn parallel to AC, AD, meeting BC, BD at F, G : shew that FG is parallel to CD.

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