Art XLVIII.—On certain Tripolar Relation: Part II.

Transactions and Proceedings of the Royal Society of New Zealand, Volume 46, 1913, Page 319

Art XLVIII.—On certain Tripolar Relation: Part II. By E. G. Hogg, M.A., F.R.A.S., Christ's College, Christchurch. [Read before the Philosophical Institute of Canterbury, 4th June, 1913.] The equation lX + mY + nZ = κ, where l + m + n =/= o, represents a system of concentric circles as κ varies. The trilinear co-ordinates (a0β0γ0) of the centre of the system are [2δl/aσ(l), 2δm/bσ(l), 2δn/cσ(l)], where δ is the area of the triangle of reference ABC. Hence the general equation of the system of concentric circles may be written aa0X+bβ0Y+cγ0Z+κ. Let (X0Y0Z0) be the tripolar co-ordinates of the centre O, and let ρ be the radius of a circle of the system. Also let this circle meet OA in the point P, and let the angles BPC, CPA, APB be called A', B', C' respectively. Then AP = OA − ρ, BP2 + ρ2 − 2OB ρ cos C' and CP2 = OC2 + ρ2 − 2OC ρ cos B'. Hence κ = aa0 (OA − ρ)2 + ρ (OB2 + ρ2 − 2OB ρ cos C') +Cγ0 (OC2 + ρ2 − 2OC ρ cos B') = aa0 OA +bβ0 Y0 + cγ0Z0 + 2δρ2 −2ρ (aa0 OA + bβ0 OB cos C' + cγ0 OC cos B' The coefficient of − 2° may be written OA. OB. OC (sin A' + sin B' cos C' + sin C' cos B') = O. Hence the equation of the circle is aα0 (X − X0) + bβ0 (Y − Y0) + cγ0 (Z − Z0) = 2δρ2 (1) Since lX + mY + nZ = κ represents a system of concentric circles, the minimum value of U ≡ lX + mY + nZ occurs for the circle of the system which reduces to a point-circle—i.e., the centre of the system. In this case U={2δ/σ(l)}2 [l/sin2A (m2/b2+n2/c2+2mm cos A/bc) +m/sin2B (n2/c2+l2/a2+2nl cos B/ca)+n/sin2C (l2/a2+2lm cos C/ab)] ={1/σ(l)}2 [lmn σ(a2) + a2mn (m+n) + b2nl (n+l)+c2lm(l+m)] =a2mn+b2nl+c2lm/σ(l) =abc(aβ0γ0+bγ0a0+c0aβ0)/2δ=2RS0 where S ≡ aβγ+bγa+caβ. Hence the equation of the circle of radius ρ whose circle is at (a0β0γ0) is aa0 X +bβ0 Y + cγ0 Z = 2RS0 + 2δρ0 (2)

If in the above equation [R cos A, R cos B, R cos C] be substituted for (a0β0γ0) and ρ be made equal to R, we obtain for the equation of the circle ABC a cos A X + b cos B Y + c cos B Z − abc = o (3) If the square of (3) be subtracted from the fundamental relation connecting four coplanar points, viz., σ(a2X2−2σ(bc cos A YZ) − 2 abc σ (a cos A X) + a2b2c2 = 0, (4) we have σ(a4X2)−2σ(b2c2YZ) = 0, whence (aX½+bY½+cZ½) (−aX½+bY½+cZ½) (aX½−bY½+cZ½) (aX½+bY½−cZ½)=0, a known relation connecting the distances of any point on a circle from the vertices of an inscribed triangle. The following list contains the equations of some of the more important circles connected with the triangle. 1. Centre at circumcentre, radius = ρ a cos A X + b cos B Y + c cos C Z = 4R sin A sin B sin C (R2 + ρ2). 2. Centre at orthocentre, radius = ρ tan A X + tan B Y + tan C Z = ρ2 tan A tan B tan C + 4δ In-circle of pedal triangle, ρ = 2R cos A cos B cos C tan A X + tan B Y + tan C Z = ½R2 sin 2A sin 2B sin 2C + 4δ. Polar circle, ρ2 = − 4R2 cos A cos B cos C tan A X + tan B Y + tan C Z = 2δ. 3. Centre at centroid, radius = ρ X + Y + Z = ⅓σ (a2) + 3σ2. 4. Centre at symmedian point, radius = ρ a2X+b2Y+c2Z=3a2b2c2/σ(a2) + σ(a2ρ2. 5. Centre at in-centre, radius = ρ aX+bY+cZ=(a+b+c) (2Rr+ρ2). In-circle, ρ = r aX − bY − cZ = (b + c − c) (2Rr + r2). 6. Centre at ex-centre (−111), radius = ρ aX − bY − cZ = (b + c − a) (2Rr1 − ρ2). Ex-circle, ρ = r1 aX − bY − cZ = (b + c − a) (2Rr1 − r12). 7. Circle concentric with circle I1I2I3, radius = ρ a (2R − r1) X + b (2R − r2) Y + c (2R − r3) Z = abcr + 2°ρ2. Circle I1I2I3, ρ = 2R a (2R − r1) X + b (2R − r2) Y + c (2R − r3) Z = abc (2R + r). 8. Centre at Nine-point centre, radius = ρ R [a cos (B − C) X + b cos (C − A) Y + c cos (A − B) Z] = R2ρ (3 + 8 cos A cos B cos C) + 4δρ2.

Nine-point circle, ρ = ½ R. a cos (B − C) X + b cos (C − A) Y + c cos (A − B) Z = abc (1 + 2 cos A cos B cos C). Orthocentric circle, ρ2 = R2/4 (1 − 8 cos A cos B cos C) a cos (B − C) X + b cos (C − A) Y + c cos (A − B) Z = abc. 9. Circles of Appolonius b2Y − c2Z = o, c2Z − a2X = o, a2X − b2Y = o. 10. Circles with centres at Brocard points Ω(c/b, a/c, b/a), Ω'(b/c,c/a,a/b), radius=ρ c2a2X+a2b2Y+b2c2Z = a2b2c2 + σ (b2c2)ρ2 b2a2X+c2b2Y+a2c2Z = a2b2c2 + σ (b2c2)ρ2 11. Circle having ΩΩ' as diameter a2 (b2 + c2) X + b2 (c2 + a2) Y + c2 (a2 + b2) Z = a2b2c2 (1 + 2 cos 2°), where ω is the Brocard angle of the triangle. 12. Brocard circle abc σ (X) + σ (a3 cos A X) = ½abc σ (a2). 13. Circle on II1 as diameter − a2X + (b + c) (bY + cZ) = a2bc. 14. Circle on I2I3 as diameter a2X − (b − c) (bY − cZ) = a2bc. 15. Circle having the bisector of the angle A as diameter (b + c) X + bY + cZ = bc (b + c). 16. Circle having side of pedal triangle as diameter a2sec A cos (B − C) X + b2 + c2Z = 8 δ2. 17. Circle on BC containing angle λ a cos λ X − b cos (c + λ) Y − c cos (B + λ) Z = abc cos (A − λ). 18. Polar circles of the triangles BOC, COA, AOB, where O is the orthocentre of the triangle ABC X = 2bc cos A, Y = 2ca cos B, Z = 2ab cos C. The tripolar equation of the Nine-point circle may be obtained in the following manner. Let X', Y', Z' be the squares of the distances of any point in the plane of the triangle ABC from the mid-points of the sides of that triangle. The Nine-point circle being the circumcircle of the medial triangle, equation is a/2 cos A X' +b/2 cos B Y' + c/2 cos C Z' = abc/8 Also Y+Z=2X'+a2/2, Z+X=2Y'+b2/2, and X +Y =2Z'+c2/2 Hence σ {a cos A (Y+Z)} =½ {abc + σ(a3 cos a)} from which we obtain σ {a cos (B−C) X} = abc (1+2 cos A cos b cos C)

If two circles of radii R + ρ and R − ρ be described concentric with the circumcircle of the triangle ABC, the tripolar equation of the pair of circles is a√X−ρ2+b√Y−ρ2+c√Z−ρ2 = 0. To prove this, let any point P be taken on the circumcircle: If λ + μ + ν = o, the co-ordinates of P may be taken to be 2δ/h(a/λ, b/μ, c/ν) where h = a2/λ+b2/μ+c2/ν. The equation of the circle of radius ρ having its centre at P will be a2/λ X + b2/μ Y + c2/ν Z = ρ2 (a2/λ + b2/μ + c2/ν) i.e., a2/λ (X − ρ2) + b2/μ (Y − ρ2) + c2/ν (Z − ρ2) = 0 and the envelope of this circle as λ, μ, ν vary is a√X−ρ2 + b√Y−ρ2 + c√Z−ρ2 = 0 (5) and this envelope consists of two circles of radii R + ρ and R − ρ concentric with the circumcircle. This equation (5) when expanded is σ (a4X2) − 2σ (b2c2YZ) + 4abc ρ2 σ (a cos A X) − 16 σ2ρ4 =0. Subtracting this from the fundamental relation (4) multiplied by 4R2 we have 4R2{σ (a2 cos2 AX2) + 2σ (bc cos B cos C YZ} − abc (2R2+ρ2) σ (a cos A X) + a2b2c2R2 + 4δ2ρ4 = 0 i.e., {σ (a cos A X)}2 − 2θ (2R2+ρ2) σ (a cos A X)+ θ2 (4R4+ρ4) = 0 hence {σ (a cos A X) − θ (2R2+ρ2)}2 = 4R2ρ2θ2, or σ (a cos A X) = θ (R2+(R±ρ)2], giving a cos A X + b cos B Y + c cos C Z = 4R sin A sin B sin C [R2 + (R ± ρ)2], a result in accord with that previously given. In a similar manner by supposing the centre of a circle of radius ρ to move on the line la+mβ+nγ = 0 we obtain the equation of the pair of straight lines parallel to la+mβ+nγ = 0 and distant ρ from it in the form l2/a2(F2−4YZ)+m2/b2(G2−4ZX)+n2/c2(H2−4XY) − 2mn/bc (GH-FX) − 2nl/ca(HF-GY) − 2lm/ab(FG-HZ)=0, where F≡Y+Z−a2−2ρ2 G≡Z+X−b2−2ρ2 H≡X+Y−c2−2ρ2

Circles of fixed radius r, whose centres are at the extremities of chords of the circle ABC passing through the symmedian point of that triangle, intersect on a circle whose centre is at the symmedian point, and whose radius is √r2−R2 tan2ω, where ω is the Brocard angle of the triangle ABC. Consider the two points P, Q, whose trilinear ratios are (a/λ, b/μ, c/v') and (a/λ', b/μ', c/v') respectively where λ + μ + v = o, and λ' = μ—v, μ' = v—λ, v' = λ—μ. The two points satisfy the equation of the circle ABC—viz., a/a + b/β + c/γ = o, and the equation of the chord PQ, λλ a/a + μμ' β/b +vv' γ/c = o is satisfied by the co-ordinates of the symmedian point (a, b, c). Hence the tripolar equations of the two circles of radius r whose centres are at P, Q are a2/λ(X−r2)+b2/μ(Y−r2)+c2/ν(Z−r2) =0 a2/λ'(X−r2)+b2/μ'(Y−r2)+c2/ν'(Z−r2) =0 Hence a2(X−r2):b2(Y−r2):c2(Z−r2) = 1/μν' − 1/μ'ν: 1/νλ − 1/ν'λ: 1/λμ' − 1/λ'μ = λλ': μμ': νν'. Therefore a2(X−r2)+b2(Y−r2)+c2(Z−r2) =0 or a2X+b2Y+c2Z = r2σ(a2), which is the equation of a circle having its centre at the symmedian point of the triangle ABC. The equation of a circle of radius r' having its centre at the symmedian point is κσ(a2X)=κ26Rabc+2δr'2 where κσ(a2)=2δ Hence σ(a2r2=κ6Rabc + 2δr'2/κ =48δ2R2/σ9a2 + σ (a2)r'2 Writing for σ (a2) its value 4δ cot ω, where ω is the Brocard angle, we have r2—r'2 = 3R2 tan2ω i.e., r' = √r2—3R2 tan2ω It follows from the above that the length of the minimum chord through the symmedian point is 2 √3 R tan ω.

If H be the circumcentre, O the orthocentre, I the in-centre, and I1 the ex-centre opposite to the vertex A of the triangle ABC, then (1) if IH = IO either A or B or C is 60°, and (2) if I1H = I1O either A is 60° or B or C is 120° (1.) The tripolar equation of a circle of radius p concentric with the in-circle is aX + bY + cZ = (a + b + c) (2Rr + p2). If this passes through H, X = Y = Z = R2; hence 2Rr + p2 = R2 and the above equation becomes aX + bY + cZ = 2R2s. If this circle passes through O, then X = 4R2 cos2A, Y = 4R2 cos2B, Z = 4R2 cos2C, and we obtain a cos2A + b cos2B + c cos2C = ½s 4a (1 − sin2A) + 4b (1 − sin2B) + 4c (1 − sin2C) = a + b + c i.e., sin 3A + sin 3B + sin 3C = o i.e, cos 3A/2 cos 3B/2 cos 3C/3 = o whence A or B or C is 60°. (2.) The tripolar equation of the circle of radius p concentric with the ex-circle opposite A is −aX + bY + cZ = (b + c − a) (p2 − 2Rr1), which, if it passes through H, reduces to − aX + bY + cZ = 2 (s − a) R2. Expressing that this circle passes through O we have − a cos2A + b cos2B + c cos2C = ½ (s − a) whence − sin 3A + sin 3B + sin 3C = o i.e., cos 3A/2 sin 3B/2 sin 3C/2 = o, and therefore either A is 60° or B or C is 120°.