Planar Triangles Research Articles

Society of doctors at Kazan University. Rhino-laryngo-otiatric section. 20th meeting 9 / XI - 1928

Dr. D. H. Matveev and engineer N. I. Pautkin. - "Mathematical definition of the approach to the antrum of the mastoid process." The outer surface of the temporal bone with the plane applied to it is always in contact at three points. One of the points lies on the zygomatic process, the other on the temporal line and the third on the most prominent part of the mastoid process. These points define a triangle called the temporal. The results of measurements of more than 50 bone preparations give the right to state the following position: the antrum of the mastoid process is in the direction of the perpendicular restored to the plane of the temporal triangle at the point of intersection of its bisectors. On the available preparations (including those with an unusual location of the antrum), the indicated perpendicular enters the temporal bone in the area of ​​the spina suprameatum Henle and approaches the antrum. In addition, being guided by the found position, an operation was performed on 40 bones. In all cases (including those with an unusual location of the antrum), they inevitably fell into the antrum without any complications in terms of damage to adjacent parts. Based on the studies performed, the following conclusion can be made: 1. The proposed approach to the antrum using the temporal triangle is expedient due to its accuracy, safety in relation to complications and the shortest length of its path. 2. The found dependence of the location of the antrum on the external parts of the temporal bone makes it possible to navigate during antrotomy according to mathematical "data". 3. The mathematical definition of the approach to the antrum raises the question of the practical use of this definition in antrotomy. - Drs. BS Goland, BN Lebedevsky, NK Trutnev, SP Yakhontov took part in the debate and prof. V.K. Trutnev.

Some Lines and Curves Associated with the Plane Triangle

Some Lines and Curves Associated with the Plane Triangle

Some Lines and Curves Associated with the Plane Triangle

Some Lines and Curves Associated with the Plane Triangle

Cartesian Equations of Circles Connected with a Plane Triangle

Cartesian Equations of Circles Connected with a Plane Triangle

Cartesian Equations of Circles Connected with a Plane Triangle

Cartesian Equations of Circles Connected with a Plane Triangle

A Generalization of Feuerbach's Theorem

It is well known that in non-euclidean geometry a plane triangle has four circumcircles, and that each of these circles touches each of four other circles. The latter theorem, which is an extension of that of Feuerbach, is essentially due to Hart.

On Pedal Quadrics in Non-Euclidean Hyperspace

1. The following are well-known theorems of elementary geometry: Given any euclidean plane triangle, A0 A1 A2, and any pair of points, X, Y, isogonally conjugate q. A0 A1 A2; then the orthogonal projections of X, Y on the sides of A0 A1 A2 lie on a circle, the pedal circle of the point-pair. If either of the points X,- Y describe a (straight) line, m, then the other describes a conic circumscribing A0 A1 A2, and the pedal circle remains orthogonal to a fixed circle, J; thus the pedal circles in question are members of an ∞2 linear system of circles of which the circle J and the line at infinity constitute the Jacobian. In particular, if the line m meet Aj Ak at Lt (i, j, k = 0, 1, 2), then the circles on Ai Li as diameter, which are the pedal circles of the point-pairs Ai, Li, are coaxial; the remaining circles of the coaxial system being the director circles of the conics, inscribed in the triangle A0 A1 A2, which touch the line m. If Mi denote the orthogonal projection on m of Ai, and Ni the orthogonal projection on Aj Ak of Mi, then the three lines Mi Ni meet at a point (Neuberg's theorem), viz. the centre of the circle J. Analogues for three-dimensional space of most of these theorems are also known ‖.

Discussions: A Check Formula for the Ambiguous Case in Plane Triangles

Discussions: A Check Formula for the Ambiguous Case in Plane Triangles

A Theorem of Sonine in Bessel Functions, with two Extensions to Spherical Harmonics

In his notable memoir on the Bessel Functions, Sonine proves the elegant result thatm > –½, provided a plane triangle can be drawn with sides a, b, c; otherwise the value of the integral is zero.

On the History of a Notation in Trigonometry

THE interpretation of formulas in the trigonometry of plane and spherical triangles is greatly simplified by the expedient of designating the sides by the same letters, respectively, as the angles opposite those sides—one group of letters being capitals, A, B, C, and the other group small letters, a, b, c. The introduction of this notation is ascribed by Moritz Cantor, in his “Geschichte der Mathematik,” vol. iii., 1901, p. 561, to the Swiss mathematician, Leonhard Euler, who first used it in 1753 (“Histoire de l'Academie de Berlin, Annee,” 1753, p. 231). It is the purpose of this note to point out that this simple, yet important, innovation was made nearly a century earlier.

A Theorem on the Contact of Circles leading up to the Theorems of Feuerbach and Hart

According to Feuerbach's theorem, the inscribed and the three escribed circles of a plane triangle are all touched by a circleHart's Theorem extends the proposition to a spherical triangle, or, which comes to the same thing, to a plane triangle formed by three circular arcs.

SOME PROPERTIES OF RATIONAL PLANE TRIANGLES WHOSE SIDES ARE WHOLE NUMBERS

School Science and MathematicsVolume 13, Issue 4 p. 320-326 SOME PROPERTIES OF RATIONAL PLANE TRIANGLES WHOSE SIDES ARE WHOLE NUMBERS Artemas Martin LL. D., Artemas Martin LL. D. Washington, D. C.Search for more papers by this author Artemas Martin LL. D., Artemas Martin LL. D. Washington, D. C.Search for more papers by this author First published: April 1913 https://doi.org/10.1111/j.1949-8594.1913.tb07718.xAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume13, Issue4April 1913Pages 320-326 RelatedInformation

The Elements of Non-Euclidean Geometry

THIS work will be found really valuable by all students of geometry, especially by those who know little or nothing of the non-Euclidean theories. First of all we have a discussion of the elementary axioms; in this the plane is deduced from what may be called a triangular frame, in the manner of Peano and Schur, Then comes the discrimination of the three cases;, according as the sum of the angles of a plane triangle is equal to, greater than, or less than two right angles; and this is followed by the fundamental trigonometric formulæ for a triangle, deduced very neatly from Saccheri's isosceles birectan-gular quadrilateral. It is also proved at this stage that the non-Euclidean plane can be developed upon a surface of constant curvature in Euclidean space. The Elements of Non-Euclidean Geometry. By Dr. J. L. Coolidge. Pp. 292. (Oxford: Clarendon Press, 1909.) Price 15S. net.

On the Connection between the Inscribed and Escribed Circles of a Triangle

The chief object of this paper is to investigate a method of obtaining from any proposition connected with the inscribed circle of a plane triangle a corresponding proposition connected with one of the escribed circles of a plane triangle.If we let a, b, c represent three lengths and A, B, C three angles, with the following limitations:I. All six quantities must be positive,II. A + B + C = π,III. we can regard a, b, c, A, B, C as the sides and angles of a plane triangle, and then by the methods of Algebra and Trigonometry we can deduce a number of algebraical and trigonometrical relations connecting them.

Note on a Variable Seven-points Circle, analogous to the Brocard Circle of a Plane Triangle

Proceedings of the London Mathematical SocietyVolume s1-25, Issue 1 p. 75-85 Articles Note on a Variable Seven-points Circle, analogous to the Brocard Circle of a Plane Triangle John Griffiths M.A., John Griffiths M.A.Search for more papers by this author John Griffiths M.A., John Griffiths M.A.Search for more papers by this author First published: November 1893 https://doi.org/10.1112/plms/s1-25.1.75AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Volumes1-25, Issue1November 1893Pages 75-85 RelatedInformation

Note on Four Special Circles of Inversion of a System of Generalized Brocard Circles of a Plane Triangle

Proceedings of the London Mathematical SocietyVolume s1-25, Issue 1 p. 376-388 Articles Note on Four Special Circles of Inversion of a System of Generalized Brocard Circles of a Plane Triangle John Griffiths M.A., John Griffiths M.A.Search for more papers by this author John Griffiths M.A., John Griffiths M.A.Search for more papers by this author First published: November 1893 https://doi.org/10.1112/plms/s1-25.1.376AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Volumes1-25, Issue1November 1893Pages 376-388 RelatedInformation

A case of non-euclidian geometry

T H E note here presented was suggested by the very interesting article by Dr. McClintock in the November number (vol. ii., p. 21) of the BULLETIN, and merely presents in another point of view the pseudo-measurements, occurring in the Cayley interpretation in the case in which c is positive, developed by Dr. McClintock (page 27 et seq.) under the name of projective distances as distinguished from the real distance of Euclidian geometry. In the theory of projective metrics, the case is the symmetric one of the opposite variety from that of Lobatschewsky, or that in which the absolute quadric of Cayley is an imaginary sphere. In the explanation given in the BULLETIN it will be remembered that, for the geometry of the plane, we have only to assume a central 'point of the plane, and a sphere of fixed radius touching it at that point ; then for any finite straight line or sect in the plane (not simply a sect of a line passing through the central point) the projective measure is the length of the arc of a great circle which is the central projection of the sect upon the spherical surface. In like manner the projective measure of an angle is the spherical angle between the great circles which are the projections upon the sphere of the sides of the given ancle. Thus a plane triangle is represented by a certain spherical triangle, and we develop a non-euclidian geometry of two dimensions, of which the theorems are those of spherical geometry. What the non-euclidian and two-dimensional dwellers in the plane would call the same triangle in different positions would be such triangles as had for projections the same spherical triangle in different positions upon the sphere. In extending these projective measures to geometry of three dimensions, we seek to establish, with reference to a single assumed central point in space, a consistent system of projective measurement for all lines and angles however situated in space. In doing this we shall establish for every plane in space its central point, and with reference to this point a system of projective measurement in the plane similar to but not identical with that described above. For this purpose Dr. McClintock distinguishes a prime plane in which the projective measurement is defined by means of a tangent sphere whose radius is r, and a secondary plane in

Early History of the Symmedian Point

In 1873, at the Lyons meeting of the French Association for the Advancement of the Sciences, Monsieur Emile Lemoine called attention to a particular point within a plane triangle which he called the centre of antiparallel medians. Since that time the properties of this remarkable point and of the lines and circles connected with it have been investigated by various writers, foremost among whom is Monsieur Lemoine himself. The results obtained by them are so numerous (indeed every month adds to their number) and so widely scattered through the mathematical periodicals of the world that it would be a task of considerable magnitude to make even an undigested collection of them. It is the purpose of the present paper to state those properties of the point which had been discovered previously to 1873. A short sketch of some of them will be found at the end of a memoir read by Monsieur Lemoine at the Grenoble meeting (1885) of the French Association, and in a memoir by Monsieur Emile Vigarie at the Paris meeting (1889) of the same Association. The references given by Dr Emmerich in his Die Brocardschen Gebilde (1891) are very valuable. It is a pity they are not more explicit.

Book Reviews

THIS small book contains a fair amount of the calculus put together in a clears and readable form. It merely touches the subject, but appears to contain enough to meet the wants of a South Kensington examinee. “It assumes a knowledge of elementary algebra and trigonometry as far as the properties of plane triangles.” The student is supposed to be unacquainted with analytical geometry, but as he is credited with a knowledge of the exponential and binomial theorems, with “indeterminate coefficients” and a few other matters, it will be seen that elementary includes a fair grasp of the two subjects named. Six chapters are devoted to the elements, successive differentiation, the theorems of Leibnitz, Taylor, and Maclaurin, maxima and minima values of a function of one variable, and the evaluation of indeterminate expressions; the remaining four chapters are devoted to elementary integration, formulæ of reduction, rational fractions, and a few applications of the integral calculus. We presume that the miscellaneous examples are taken from South Kensington papers; those in the text are old friends which figure in Todhunter's works. In the text, the following slips occur: p. 4, 1. 15, for f(x) read f'(.r); p. 18 (6), read exx; p. 37 (3), for x6 720 we get x6; p. 40 (3),? (a - b)2;2la for the maximum; p. 41, 1. 4 up, for 2^+3^, read 3^ + 2^; p. 42 (i), read cos3 6 and 3 \f$d2/i6; p. 62 (4),? last connecting sign (read -); p. 71 (4), for TT read 7T2; p. 80 (24). in first place read (i+^r2)2. In the answers, we differ from the author in (i), (20), (74), and (88). We prefer to work (84) from f (i +/a)V/, where t stands for tan x.

Book Reviews

THIS small book contains a fair amount of the calculus put together in a clears and readable form. It merely touches the subject, but appears to contain enough to meet the wants of a South Kensington examinee. “It assumes a knowledge of elementary algebra and trigonometry as far as the properties of plane triangles.” The student is supposed to be unacquainted with analytical geometry, but as he is credited with a knowledge of the exponential and binomial theorems, with “indeterminate coefficients” and a few other matters, it will be seen that elementary includes a fair grasp of the two subjects named. Six chapters are devoted to the elements, successive differentiation, the theorems of Leibnitz, Taylor, and Maclaurin, maxima and minima values of a function of one variable, and the evaluation of indeterminate expressions; the remaining four chapters are devoted to elementary integration, formulæ of reduction, rational fractions, and a few applications of the integral calculus. We presume that the miscellaneous examples are taken from South Kensington papers; those in the text are old friends which figure in Todhunter's works. In the text, the following slips occur: p. 4, 1. 15, for f(x) read f'(.r); p. 18 (6), read exx; p. 37 (3), for x6 720 we get x6; p. 40 (3),? (a - b)2;2la for the maximum; p. 41, 1. 4 up, for 2^+3^, read 3^ + 2^; p. 42 (i), read cos3 6 and 3 \f$d2/i6; p. 62 (4),? last connecting sign (read -); p. 71 (4), for TT read 7T2; p. 80 (24). in first place read (i+^r2)2. In the answers, we differ from the author in (i), (20), (74), and (88). We prefer to work (84) from f (i +/a)V/, where t stands for tan x.