Ex. 181. ABCD is a quadrilateral with the angles at A and C right angles. If BK and DN are drawn perpendicular to AC, prove that AN=CK.

Ex. 132. In fig. 127, PQ is drawn parallel to AD to meet BC produced in Q; prove that PQ is a mean proportional between QB, QC.

Ex. 133. ABC is a triangle right-angled at A; AD is drawn perpendicular to BC and produced to E so that DE is a third proportional to AD, DB; prove that ▲ ABD ACDE, and that ▲ ABD is a mean proportional between ▲ 8 ADC, BDE.

Ex. 184. Two circles touch externally at P; Q, R are the points of contact of one of their common tangents. Prove that QR is a mean proportional between their diameters. [Draw the common tangent at P, let it cut QR at S; join S to the centres of the two circles.]

Ex. 135. Two circles ACB, ADB intersect at A, B; AC, AD touch the circles ADB, ACB respectively at A; prove that AB is a mean proportional between BC and BD.

Ex. 136. ABCD is a parallelogram, P is a point in AC produced; BC, BA are produced to cut the straight line through P and D in Q, R respectively. Prove that PD is a mean proportional between PQ and PR.

Ex. 137. The sides AB, AC of a triangle are bisected at D and E respectively; prove that, if the circle ADE intersect the line BC, and P be a point of intersection, then AP is a mean proportional between BP and CP.

CHAPTER XIII

AN INTRODUCTION TO MORE ADVANCED GEOMETRY*

GEOMETRY OF THE TRIANGLE.

In Exx. 1-33 the triangle referred to is ▲ ABC unless otherwise stated.

Ex. 1. The perpendicular bisectors of the sides of a triangle are concurrent; and the point of concurrence, S, is the circumcentre.

Ex. 2. The internal bisectors of the angles of a triangle are concurrent; and the point of concurrence, I, is the in-centre.

Ex. 3. The internal bisector of LA and the external bisectors of Ls B and C are concurrent; and the point of concurrence is the ex-centre |1.

Ex. 4. The medians of a triangle are concurrent, and the point of concurrence is a point of trisection of each median; G, the point of concurrence, is called the centroid. Prove this (i) by similar triangles, (ii) by drawing medians from B and C to intersect at G, producing AG to P so that GP=AG, and then proving GBPC a parallelogram.

Ex. 5. The altitudes BE, CF of A ABC intersect at H; prove

(i) that AEHF is a cyclic quadrilateral,

(iv) that, if AH is produced to meet BC in D, AFDC is cyclic,

(v) that AD is to CB.

Hence the three altitudes of a triangle meet in a point, which is called the orthocentre.

Ex. 6. If the sides of a triangle are a, b, c; 28=a+b+c; A= area of triangle; radius of in-circle=r, of ex-circle opposite to A=r1; then r=A/s, r1 =▲/(s − a). [Use A ABC ABIC + ▲ CIA + A AIB

and Δ ΑΒC = Δ ΑΙΒ+ Δ ΑΙ C - ΔΒΙ, C.]

Ex. 7. If the sides of the triangle ABC touch the in-circle at X, Y, Z and the escribed circle opposite A in X1, Y1, Z1, prove that AY1 =AZ1 = AY=AZ=8-a.

=s and

Ex. 8. BE, CF, two altitudes of ▲ ABC, intersect at H. BE produced meets the circumcircle in K. Prove that E is the mid-point of HK.

[Show that BFEC is a cyclic quadrilateral, .. LFCE=4FBE.

But LKCE=LKBF, .. etc.]

* For a fuller treatment see Godfrey and Siddons' Modern Geometry.

Ex. 9. If AD, BE, CF are the altitudes of a triangle ABC and H its orthocentre, DEF is called its pedal triangle. Prove that (i) each of the angles EDC, FDB, EHC, FHB is equal to LA, (ii) H is the in-centre of ▲ DEF.

Ex. 10. ABC, A'CB are two congruent triangles on the same side of the base BC. Prove that A, B, C, A' are concyclic.

Hence show that if D, E, F are the mid-points of the sides BC, CA, AB of ▲ ABC, and AL is an altitude, then D, E, F, L are concyclic.

Hence, the circle through the mid-points of the sides of a triangle also passes through the feet of the altitudes.

Ex. 11. In fig. 137, AD is 1 to BC and BE is to CA; S is the centre of the circle. Show that BF = AH, and that AB, FH bisect one another. [Prove AHBF & parallelogram.]

Ex. 12. If a is the mid-point of BC, AH=2Sa (fig. 137).

Ex. 13. A circle whose centre is the mid-point of SH, and whose radius is R, passes through D, E, F, the feet of the altitudes, a, ß, y, the mid-points of the sides, P, Q, R, the mid-points of HA, HB, HC. (The nine-points circle.)

Ex. 14. The points H, G, S are collinear; and HG=2GS.

Ex. 15. In fig. 138, P is any point on circumcircle of ▲ ABC. PL, PM, PN are i to BC, CA, AB respectively. Prove that (i) PNL=180° - 4PBC, (ii) 4PNM=LPAM, (iii) 4PNL+2PNM=180°, (iv) LNM is a straight line.

Verify this result by drawing. LNM is called Simson's line.

D

ΔΕ

Fig. 137.

Fig. 138.

Ex. 16. The altitude from A is produced to meet the circumcircle in X, and X is joined to any point P on the circumcircle. PX meets the Simson line of P in R; and BC in Q. Prove that R is the mid-point of PQ.

Ex. 17. In Ex. 16 show that HQ is parallel to the Simson line of P. (H is the orthocentre of ▲ ABC.)

Ex. 18. Deduce, from Ex. 17, that the line joining a point on the circumcircle to the orthocentre is bisected by the Simson line of the point.

Ex. 19. If the lines joining a point O to the vertices of a triangle ABC meet the opposite sides in X, Y, Z, then

BX CY AZ
CX AY BZ

= 1, the sense of lines

being taken into account. (Ceva's theorem; 1678 A.D.)

[Use areas.]

E

D

Ex. 20. If a straight line cuts the sides of a triangle ABC in L, M, N, then BL CM AN

+1, the sense of lines being taken into account. (Menelaus'

theorem; about 100 A.D.)

[Draw perpendiculars from the vertices.]

Ex. 21. State and prove the converse of Ceva's theorem.

Ex. 22. State and prove the converse of Menelaus' theorem.

Ex. 23. Use Ceva or its converse (be careful to state which you are using) to prove the concurrence of

(i) the medians of a triangle;

(ii) the bisectors of its angles;

(iii) its altitudes;

(iv) the bisectors of one interior and the two other exterior angles.

Ex. 24. The bisectors of the Ls B and C meet the opposite sides in Q, R, and QR meets BC in P; prove that AP is the exterior bisector of LA.

Ex. 25. a, ß, y are the mid-points of the sides; Aa meets By in P; CP meets AB in Q. Show that AQ=AB.

Ex. 26. I is the centre of the inscribed circle of A ABC; I, is the centre of the circle escribed outside BC. Prove that BICI1 is cyclic.

Ex. 27. An escribed circle of AABC touches BC externally at D, and touches AB, AC produced at F, E respectively; O is the centre of the circle.

Hence find the locus of the inscribed centre of a triangle whose base and vertical angle are given.

Ex. 29. is the centre of the inscribed circle of ▲ ABC; Al produced meets the circumcircle in P; prove that PB=PC=PI.

Ex. 30. If the vertical angle A of a triangle ABC is bisected by a straight line which cuts the base in D and the circumcircle of the triangle in E, prove that

AB. AC AD.AE=AD2+BD. DC.

Hence find an expression in terms of a, b, c for the length of the bisector AD.

Ex. 31. ABC is a triangle, AD the altitude from A and AE the diameter of the circumcircle through A. Prove that

AB.AC=AE. AD.

Hence prove that the radius of the circumcircle=

abc
4A'

where A is the area

of the triangle ABC.

Ex. 32. If p, q, r be the perpendicular distances of the vertices of a triangle from any straight line, prove that the distance of the centroid is (p+q+r). Ex. 33. If H is the orthocentre, prove that

AH2+BC2=BH2+ CA2=CH2+ AB2=d2,

where d is the diameter of the circumcircle.

HARMONIC SECTION.

Ex. 34. Show how to divide a line, by geometric construction, internally and externally in the same given ratio. (Such a line is said to be divided harmonically.)

Ex. 35. Prove that the internal and external bisectors of an angle of a triangle divide the opposite side harmonically.

Ex. 36. The line AB is divided harmonically at C and D; prove that

[Use symbols for lengths measured from A.]

Ex. 37. In Ex. 36 O is the mid-point of AB; prove that OC. OD=OB2. [Use symbols for lengths measured from O.]

Ex. 38. The tangents at the extremities of a chord HK intersect at T. Prove that any line from T is divided harmonically by the circle and the chord HK.

THE CIRCLE OF APOLLONIUS*.

Ex. 39. A point P moves so that the ratio of its distances from two fixed points Q, R is constant; prove that the locus of P is a circle. [Draw the internal and external bisectors of LP.]

Ex. 40. Construct a triangle ABC, in which BC is 2", the angle BAC is 60°, and the sides AB and AC are in the ratio 3: 4.

Ex. 41. Show how to construct a triangle having given the vertical angle, the ratio of the sides containing the angle, and the altitude drawn to the base.

Ex. 42. Two church spires stand on a level plain; a man walks on the plain so that he always sees the tops of the spires at equal angles of elevation. Prove that his locus is a circle.

ORTHOGONAL CIRCLES.

Orthogonal Circles are two circles which intersect at right angles: that is, their tangents at a point of intersection are at right angles.

Ex. 43. If two circles are orthogonal, a tangent to either at their point of intersection passes through the centre of the other.

Ex. 44. The sum of the squares of the radii of two orthogonal circles is equal to the square on the distance between their centres.

* Apollonius' circle can be used in Practical Geometry to solve many problems on relative motion.