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ABCD AC is equal angle ABC angle BAC angle DAE angle DEF angle HBK angular points antecedent base chord HK circle ABC circle whose centre circles touch circumscribed circle coincide decagon diagonal diameter divided internally draw duplicate ratio equiangular equimultiples externally given circle given point given ratio given straight line greater half the angle Hence homologous sides inscribed circle isosceles triangle less Let ABC line joining mean proportional meet the circumference middle point minor arc HK multiple nine-points circle orthocentre parallel parallelogram perpendicular Plane Geometry point of contact polygon Prob Prove Q.E.D. Theor quadrilateral radii radius ratio compounded ratios are equal rectangle BD rectangle contained rectilineal figure right angles sector segment BAC semicircle Shew side BC square straight line drawn tangent triangle ABC triangle DEF vertical angle
Page 90 - CD are three sides of a regular pentagon inscribed in the circle OBC. Ex. 156. Describe an isosceles triangle having each of the angles at the base one-third of the vertical angle. •• : £x. 157. Divide a right angle into five equal parts. Ex. 158. Shew that each diagonal of a
Page 71 - therefore the triangle ABC is equiangular to the triangle DEF, and it is inscribed in the circle ABC. DEF. 17. If all the sides of a rectilineal figure touch a circle lying within the figure, the circlets said to be inscribed in the figure, and the figure to be circumscribed about the circle.
Page 126 - B : Aa ratio of less inequality. Also the ratios A : B and B : A are said to be reciprocal to one another. THEOR. 1. Ratios that are equal to the same ratio are equal to one another. Let A : B :: P : Q and also
Page 111 - D + F). Props, of Mults. 3. Hence A : B : : A + C + E :B + D + F. Def. 5. or, if any number of magnitudes of the same kind be proportionals, as one of the antecedents is to its consequent, so is the sum of the antecedents to the sum of the consequents. QED COR. If A : B : : C : D (A being greater than C), then
Page 70 - 15. The circle that touches the three sides of a triangle is called the inscribed circle of the triangle. DEF. 16. A circle that touches one side of a triangle and the other two sides produced is called an escribed circle of the triangle.
Page 114 - =».FH. Hence AC : EG : : BD : FH. Def. 5. QED COR. 1. If there are three parallel straight lines, the intercepts made by them on any straight line that cuts them, are to one another in the same ratio as the
Page 83 - 28. If a chord of a circle is divided into two segments by a point in the chord or in the chord produced, the rectangle contained by these segments is equal to the difference of the squares on the radius and on the line joining the given point with the centre of the circle. Let
Page 148 - equal to the second ratio, and so on, then the first of the set is said to have to the last the ratio compounded of the original ratios. DEF. 11. When two ratios are equal, the ratio compounded of them is called the duplicate ratio of either of
Page 52 - 16, Cor. 1. QED COR. 1. The two tangents drawn to a circle from an external, point are equal, and make equal angles with the straight line joining that point and the centre. / 20, Cor. 1. COR. 2.
Page 137 - DEF. 10. If there are any number of ratios, and a set of magnitudes is taken such that the ratio of the first to the second is equal to the first -ratio, and the ratio of the second to the third is equal to the second ratio, and so on, then the first of the set is said to