By Julian Lowell Coolidge
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Extra resources for A History of Geometrical Methods
With the A ABC ratio is % A* and so forth. Continuing thus we have in (AB) and (AC) two infinite series of points. Let the reader show that the A limit for each series cannot be other than the point itself. 4, hence Now a sin r = b . sin j A/ A/ Let the reader deduce from tan r (4) ,. M and (5) (5) that = tan r cos A. A/ cos . sin A. (6) A/ B= cos f sin A. A* (7) ABC If none is any triangle. Let us next suppose that A of the angles be greater than a right angle, we may connect any vertex with a point of the opposite side by a line perpendicular to the line of that side, and we see at once that .
An exterior angle of a triangle is comparable with either of the opposite interior angles. Let us take the triangle ABG, while D lies on the extension of (BC) beyond C, Let E be the middle point of ( AG) and let in F. If ~DE > RF find of (DE) so BAG congruent to 4-ECG Then we have Df^meet^AB) that FE = EG. * Cf. Borel, Lemons sur la theorie de$ functions, Paris, 1898, pp. 102-8. CONGRUENT TRANSFORMATIONS ii and less than 4-ECD. BAC greater than an angle congruent to 4- EGD. Theorem 27. Two angles of a triangle are comparable.
In a rectangle the opposite sides are mutually congruent, and any isosceles birectangular quadrilateral whose opposite sides are mutually congruent is necessarily a rectangle. TJteorewi 9. If there exist a single rectangle, every isosceles birectangular quadrilateral is a rectangle. be the rectangle. The line perpendicular to Let at the middle point of (AB) will divide it into two smaller rectangles. Continuing this process we see that we can construct a rectangle whose adjacent sides may have any ABCD AB _ measures that can be indicated in the form __ AB, i ~ - t < * - AC, provided, of course, that the distances so called for exist simultaneously on the sides of a birectangular isosceles Distances so indicated will be everywhere quadrilateral.
A History of Geometrical Methods by Julian Lowell Coolidge