Method in Geometry |
Other editions - View all
Common terms and phrases
ABCD Analytic Geometry angles equal assume the theorem bisect bisector BOSTON NEW YORK Bowser's called colleges and technical concyclic congruent triangles D. C. HEATH definition demonstrated diagonal discovered discovery drawing auxiliary lines drawn eleventh grade Essentials of Algebra Essentials of Geometry Euclid example exterior angle Four Place Tables fundamental given angle given points hence intersection of loci isosceles known large number lines of procedure logic Mathematics median METHOD OF ATTACK number of solutions opportunity for original opposite sides parallel original exercises parallel lines parallelogram perpendicular Plane and Spherical plane geometry problems pupil quadrilateral recall reductio ad absurdum required circle right triangles secant line segment sides equal sinx solution of right solve Spherical Triangles Spherical Trigonometry straight line student of geometry study of geometry suppose tangent teacher teaching Theory of Equations tion trapezoid triangles congruent Trigonometric Functions vertex Wells's Essentials YORK CHICAGO
Popular passages
Page 21 - ... possibly suggest a proof. He is now ready after following each suggestion to its limit, to select those which it is possible to use as well as those which it is best to use. When he has done this for some time, until he has "at his tongue's end" all propositions and corollaries that he has proven, he sees in each of the terms involved in a theorem, not only the one definition of the term but many, and is able to select the particular ones that will lead to a proof. For example, he comes to think...
Page 23 - Two triangles are equal if the three sides of the one are equal, respectively, to the three sides of the other. In the triangles ABC and A'B'C', let AB be equal to A'B', AC to A'C', BC to B'C'. To prove that A ABC = A A'B'C'. Proof. Place A A'B'C
Page 24 - substitute the definition in the place of the name of the thing denned" and say, to prove AB=DC. 4. Analysis. To prove lines equal requires, 1. congruent triangles, 2. parallelograms, etc. (a) To prove the triangles congruent requires that we have 1. The three sides of one equal, respectively, to the three sides of the other, 2. two angles and a side of one equal, respectively, to two angles and a side of the other, or 3. an angle and two sides of one equal, respectively, to an angle and two sides...
Page 32 - ... involve certain elements of the old, just as was done in the case of theorems. This might be called the method of transformation. Just as the most difficult theorems were those that had to be transformed by auxiliary lines, so will this class of problems give the most trouble, and the skill in being able to see the needed auxiliary lines will largely determine a student's success with exercises of this sort. As an example, suppose we are to construct a triangle having given two sides and the...
Page 24 - If two triangles have the three sides of one equal respectively to the three sides of the other, the triangles are congruent, (sss) REMARK 1.
Page 29 - ... touch one another, there be drawn two parallel diameters, the point of contact and an extremity of each diameter, lie in the same straight line. 3. Describe a circle which shall touch a given circle, have its centre in a given straight line, and pass through a given point in the given straight line. 4. Describe a circle of given radius to pass through a given point and to touch a given circle.
Page 38 - Conversely, if two straight lines cut one another so that the rectangle contained by the augments of the one is equal to the rectangle contained by the segments of the other, the four extremities of the two straight lines are concyclic.
Page 19 - It is also very important that the student know the definition of all terms in the theorem, and instead of the hypothesis and conclusion of the theorem it will be well to use Pascal's advice and "substitute the definition for the name of the thing defined" and thus get a new hypothesis and a new conclusion. 3. Next, recall all the propositions that can have a bearing upon the exercise under consideration. I believe that a large part ol the difficulty that a beginning student has, comes from his not...