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A Sequel to the First Six Books of the Elements of Euclid, by John Casey

By John Casey

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Extra resources for A Sequel to the First Six Books of the Elements of Euclid, Fifth Edition

Example text

S be drawn to the three sides of a � , prove that the sum of the squares of three alternate segments of the sides = the sum of squares of the three remaining segments. 20. Prove the following theorem, and infer from it Prop. xlvii . : If CQ, CP be Os described on the sides CA, CB of a � , and if the sides 11 to CA, CB be produced to meet in R, and RC joined, a D described on AB with sides = and 11 to RC shall be = to the sum of the OS CQ, CP. 2 1 . If a square be inscribed in a 6. , the rectangle under its side and the sum of base and altitude = twice the area of the � .

GF = GF' . -The line GD is called the railical axis of the two circles ,. and two points I, I', taken on the hOne through the centres, BO that DI -:- DI' = DE = DE', are called the lz·m�·ting pozOn ts. (Jor. -Any circle who s e centre is on the radical axis, and which cuts one of the given 0 s orthogonally, will also cut the oth er orthogonally, and will pass through the two limiting points. -1f there be a sy stem of three 0 s, · their radical axes taken in pairs ar e c on curr ent . F or , if t an g en ts be drawn to the 0 s from the point or int er­ section o f two or the radical axes, the three t an gents will be equal.

Prop. -If two circles touch each other at any point P, and any line out the o�'r oles �·n P the points A, B, 0, D ; then the :ID (tngle APB = OPD. ) L E PA = PDA. Hence, by subtraction, L AP13 OPD Prop. -If a oirole touoh a semioirole �'n D ana its d�'ameter in P, and PE be per­ pendioular to the diameter at P, the square on PE is equal to tw�'oe the reotang le oontaz'n ed by the radii of the oiroles. -Oomplete the circle, A and produce EP to meet it again in G. Let 0 and F be the centres ; then the line OF will pass through D .

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