Miscellaneous Propositions

Excerpts

• Miscellaneous Propositions

  *If three arcs be drawn from the angles of a spherical triangle through any point to meet the opposite sides, the products of the sines of the alternate segments of the sides are equal.*

• Miscellaneous Propositions

  *The arc which passes through the middle points of the sides of any triangle upon a given base will meet the base produced at a fixed point, the distance of which from the middle point of the base is a quadrant.*

• Miscellaneous Propositions

  It may be presumed from symmetry that the pole of this circle is in the great circle which bisects $AB$ at right angles; and this presumption is easily verified.

• Miscellaneous Propositions

  Let $P$ denote the pole of the inscribed circle, and $Q$ the pole of the circumscribed circle of a triangle $ABC$;

• Miscellaneous Propositions

  then the spherical triangle which corresponds to the three planes $LPM$, $MPN$, $NPL$ is the *polar triangle* of the spherical triangle which corresponds to the solid angle at $O$.

• Miscellaneous Propositions

  Again, we know in mechanics that if three forces acting at a point are in equilibrium, each force is as the sine of the angle between the directions of the other two: the following proposition is analogous; if four forces acting at a point are in equilibrium each force is as the sine of the solid angle formed by the directions of the other three.

• Miscellaneous Propositions

  the volume of a tetrahedron is one sixth of the product of three edges into the sine of the solid angle which they form.

• Miscellaneous Propositions

  The result, however, is generally true, even in cases in which the condition required by the demonstration of Art. 150 is not satisfied.

• Miscellaneous Propositions

  We begin with a theorem which is due to Cauchy.

• Miscellaneous Propositions

  Hence it may be inferred that any change of position in a rigid body, of which one point is fixed, may be effected by rotation round some axis through the fixed point.

Equations

• Miscellaneous Propositions

  \cos A' = \sin (S - A) \cos \frac{a}{2}

  The cosine of an angle of the chordal triangle equals the sine of S minus the corresponding angle times the cosine of half the side (example 13).

• Miscellaneous Propositions

  \dfrac{\sin Pa \cos PA}{\sin Aa} + \dfrac{\sin Pb \cos PB}{\sin Bb} + \dfrac{\sin Pc \cos PC}{\sin Cc} = 1

  For a point P inside a spherical triangle with cevian great circles through P, the three weighted sine-cosine ratios sum to one (example 16).

• Miscellaneous Propositions

  \cot \tfrac{1}{2}E = \cot \tfrac{1}{2}\theta \cot \tfrac{1}{2}c \operatorname{cosec} \phi + \cot \phi

  The cotangent of half the spherical excess equals a combination of the cotangents of half the sides and the angle at A, for a triangle with given base and area.

• Miscellaneous Propositions

  \cos \theta \cot \tfrac{1}{2}c \sin \tfrac{1}{2} E + \sin \theta \cos \left(\phi - \tfrac{1}{2}E + \dfrac{\pi}{2}\right) = -\cot \tfrac{1}{2} c \sin \tfrac{1}{2} E

  The vertex of a spherical triangle of given base and area satisfies a linear trigonometric equation in theta and phi, which shows its locus is a circle.

• Miscellaneous Propositions

  \beta = \tfrac{1}{2} E - \frac{\pi}{2}

  The angular co-ordinate beta of the pole of the locus circle equals half the spherical excess minus a right angle.

• Miscellaneous Propositions

  0 = \cos \theta \cos \left(\frac{\pi}{2} - \frac{c}{2}\right) + \sin \theta \sin \left(\frac{\pi}{2} - \frac{c}{2}\right) \cos (\phi - \pi)

  The equation of the great circle that bisects the base AB at right angles, in the polar co-ordinates theta and phi.

• Miscellaneous Propositions

  \cos PAQ = cos\tfrac{1}{2}(B-C)

  The angle between the arcs from A to the poles P and Q of the inscribed and circumscribed circles equals the cosine of half the difference of the angles B and C.

• Miscellaneous Propositions

  \cos PQ = \cos PA \cos QA + \sin PA \sin QA \cos \tfrac{1}{2}(B-C)

  The spherical cosine rule applied to the triangle with vertices P, A, Q gives the cosine of the distance PQ between the two poles.

• Miscellaneous Propositions

  \sin PA = \frac{\sin PE}{\sin PAE} = \frac{\sin r}{\sin\tfrac{1}{2}A}

  The sine of the distance PA from the vertex to the pole of the inscribed circle equals the sine of the inradius divided by the sine of half the angle A.

• Miscellaneous Propositions

  \cos PQ = \cos R \cos r \cos(s-a) + \sin R \sin r \sin \tfrac{1}{2}(b+c) \operatorname{cosec} \tfrac{1}{2}a

  The cosine of the distance between the two poles is expressed in terms of the inradius r, circumradius R, and the sides of the triangle.

• Miscellaneous Propositions

  \cot r = \frac{\sin s}{n}

  The cotangent of the inradius equals the sine of the semi-perimeter divided by n, where n is a quantity defined from the sides.

• Miscellaneous Propositions

  \tan R = \frac{2 \sin \frac{1}{2} a \sin \tfrac{1}{2} b \sin \tfrac{1}{2} c}{n}

  The tangent of the circumradius R equals twice the product of the sines of half the sides, divided by n.

• Miscellaneous Propositions

  \frac{\cos PQ}{\cos R \sin r} = \cot r \cos(s-a) + \tan R \sin \frac{1}{2}(b+c) \operatorname{cosec} \frac{1}{2}a

  Dividing the cosine formula for PQ by cos R sin r gives a form in cotangents and tangents of the radii.

• Miscellaneous Propositions

  \cos^2 PQ = \cos^2 R \sin^2 r + \cos^2 (R-r)

  The squared cosine of the distance between the inscribed-circle and circumscribed-circle poles is given in terms of the circumradius R and inradius r.

• Miscellaneous Propositions

  \sin^2 PQ = \sin^2 (R-r) - \cos^2 R \sin^2 r

  The squared sine of the distance between the poles of the inscribed and circumscribed circles is given in terms of R and r.

• Miscellaneous Propositions

  \cos QQ_1 = \cos R \cos r_1 \cos (s-c) - \sin R \sin r_1 \sin \tfrac{1}{2}(C-A) \sec \tfrac{1}{2}B

  The cosine of the distance between the circumscribed-circle pole Q and the escribed-circle pole Q1 is given by a spherical cosine relation.

• Miscellaneous Propositions

  \cos^2 QQ_1 = \cos^2 R \sin^2 r_1 + \cos^2 (R + r_1)

  The squared cosine of the distance QQ1 between the circumscribed and escribed poles in terms of R and r1.

• Miscellaneous Propositions

  \sin^2 QQ_1 = \sin^2 (R + r_1) - \cos^2 R \sin^2 r_1

  The squared sine of the distance QQ1 between the circumscribed and escribed poles in terms of R and r1.

• Miscellaneous Propositions

  \sin BQ = \sin CQ

  The sines of the arcs from B and from C to the point Q, where the arc through the midpoints meets BC produced, are equal.

• Miscellaneous Propositions

  BQ + CQ = \pi

  The arcs BQ and CQ together make a half great circle, so the fixed point Q lies a quadrant beyond the midpoint of the base.

• Miscellaneous Propositions

  DQ = \tfrac{1}{2} (BQ + CQ) = \tfrac{1}{2} \pi

  The distance from the midpoint D of BC to the fixed point Q is a quadrant.

• Miscellaneous Propositions

  \sin BD \sin CE \sin AF= \sin CD \sin AE \sin BF

  If three arcs from the vertices of a spherical triangle pass through a common point, the products of the sines of alternate segments of the sides are equal.

• Miscellaneous Propositions

  \dfrac{\sin BD}{\sin CD}\, \dfrac{\sin CE}{\sin AE}\, \dfrac{\sin AF}{\sin BF} = 1

  The product of the three sine ratios of the divided sides equals one (Ceva-type condition for concurrent arcs on a sphere).

• Miscellaneous Propositions

  \surd(1-\cos^2 \alpha-\cos^2 \beta-\cos^2 \gamma+2\cos \alpha \cos \beta \cos \gamma)

  The sine of a solid angle formed by three plane angles alpha, beta, gamma is the square root of the stated expression, which lies between zero and one.

• Miscellaneous Propositions

  2E=3a+4b+5c+6d+ \ldots\ldots

  Each edge belongs to two faces, so twice the number of edges equals the sum of sides over all faces.

• Miscellaneous Propositions

  2E= 3 \alpha + 4 \beta + 5 \gamma + 6 \delta + \ldots\ldots

  Each edge terminates at two solid angles, so twice the number of edges equals the sum of plane-angle counts over all solid angles.

• Miscellaneous Propositions

  F=a+b+c+d+ \ldots\ldots

  The total number of faces is the sum of the numbers of faces with each number of sides.

• Miscellaneous Propositions

  S= \alpha + \beta + \gamma + \delta + \ldots\ldots

  The total number of solid angles is the sum of the numbers of solid angles with each number of plane angles.

• Miscellaneous Propositions

  2E-3F = b + 2c + 3d + \ldots\ldots

  Twice the number of edges minus three times the number of faces equals a sum with nonnegative coefficients, so 2E cannot be less than 3F.

• Miscellaneous Propositions

  2E-3S = \beta + 2 \gamma + 3 \delta + \ldots\ldots

  Twice the number of edges minus three times the number of solid angles equals a sum with nonnegative coefficients, so 2E cannot be less than 3S.

• Miscellaneous Propositions

  2F + 2S=4 + 2E

  Twice the sum of faces and solid angles equals four plus twice the number of edges, which is Euler's relation rewritten.

• Miscellaneous Propositions

  2 (\alpha + \beta + \gamma + \delta + \ldots) - (a + 2b + 3c + 4d + \ldots) = 4

  Combining the face and solid-angle counts gives an identity relating solid-angle counts to face-side counts.

• Miscellaneous Propositions

  2 (a + b + c + d + \ldots) - (\alpha + 2 \beta + 3 \gamma + 4 \delta + \ldots) = 4

  The dual identity relating face counts to solid-angle counts holds for any polyhedron.

• Miscellaneous Propositions

  a + \alpha - (c + \gamma) - 2 (d + \delta) - 3 (e + \epsilon) - \ldots\ldots = 8

  Adding the two identities shows the number of triangular faces plus triangular solid angles is at least eight.

• Miscellaneous Propositions

  3a + 2b + c - e - 2f - \ldots\ldots -2\beta - 4\gamma - \ldots\ldots = 12

  Eliminating alpha gives a relation showing 3a + 2b + c cannot be less than 12.

• Miscellaneous Propositions

  \mathrm{F + S = E + 1 }

  For a network of rectilineal figures not forming a closed surface, faces plus corner points equals edges plus one.

• Miscellaneous Propositions

  F-1+S = E + 1

  Removing one face of a polyhedron yields a network to which Cauchy's theorem applies.

• Miscellaneous Propositions

  F + S = E + 2

  For any polyhedron, faces plus solid angles equals edges plus two.

• Miscellaneous Propositions

  1 + e + e' = s + s' + F

  Splitting edges, corners and faces into those on the bounding contour and those within it gives a relation for the network.

• Miscellaneous Propositions

  1 + e' = s' + F

  Since the contour edges and corners are equal in number, the interior edges and corners satisfy 1 + e' = s' + F.

• Miscellaneous Propositions

  \mathrm{S + F = E + P + 1 }

  If a polyhedron is decomposed into P polyhedrons, solid angles plus faces equals edges plus P plus one.

• Miscellaneous Propositions

  \tan c = \frac{\cot A \cot a + \cot B \cot b} {\cot a \cot b - \cos A \cos B}

  The tangent of side c of a spherical triangle is given in terms of two angles and their opposite sides (example 4).

• Miscellaneous Propositions

  \cos \theta \sin(b - c) + \cos \phi \sin(c - a) + \cos \psi \sin(a - b) = 0

  For the point where the angle bisectors of a spherical triangle meet, the stated weighted cosine sum vanishes (example 5).

• Miscellaneous Propositions

  \cos PA \cos BC = \cos PB \cos CA = \cos PC \cos AB

  For the point P where the great circles through the vertices and the poles of the opposite sides meet, the products of cosines are equal (example 6).

• Miscellaneous Propositions

  \tan \alpha \tan \alpha' = \tan \beta \tan \beta' = \tan \gamma \tan \gamma'

  For the three cevian arcs perpendicular to the sides, the products of the tangents of their segments are equal (example 8).

• Miscellaneous Propositions

  \frac{\cos p}{\cos \alpha \cos \alpha'} = \frac{\cos q}{\cos \beta \cos \beta' } = \frac{\cos r}{\cos \gamma \cos \gamma'}

  For the perpendicular arcs p, q, r of a spherical triangle, the ratios of their cosines to products of cosines of their segments are equal (example 8).

• Miscellaneous Propositions

  \frac{\sin \alpha}{\sin \alpha'} = 2 \cos \frac{a}{2}

  For an arc from a vertex to the midpoint of the opposite side, the ratio of the sines of its two parts equals twice the cosine of half that side (example 9).

• Miscellaneous Propositions

  \cos AQ \sin \frac{a}{2} = \sin \frac{c - b}{2} \sin \frac{c + b}{2}

  For the arc bisecting AB and AC meeting BC produced at Q, the cosine of AQ times the sine of half a equals a product of sines (example 10).

• Miscellaneous Propositions

  \sin AB \sin CD \cos P = \sin AD \sin BC \cos Q = \sin AC \sin BD \cos R

  For a spherical quadrilateral with the stated intersection points, the three products of sines and cosines are equal (example 12).

• Miscellaneous Propositions

  a \cos AP + b \cos BP = s \cos SP

  For fixed points A and B on a sphere and constants a and b, a fixed point S exists on AB such that the combined cosine relation holds for every P (example 19).

• Miscellaneous Propositions

  a \cos AP + b \cos BP + c \cos CP + \ldots = \text{constant}

  For fixed points A, B, C, ... on a sphere with given constants, the locus of P satisfying a constant weighted cosine sum is a circle (example 20).

Problems