...restriction is plain, when we consider that he had previously proved in his Prop. XX., Book I. that any two sides of a triangle are together greater than the third. This, however, is manifest without demonstration, if we adopt Archimedes's definición of a straight...

...each ? 5. If a : b : : c : d show that a + b : J : : c + d : d and that am : b" : : C" : d" EUCLID. 1. Any two sides of a triangle are together greater than the third side. the squares of the sides containing it by twice the rectangle contained by the side upon which, when...

...line. What figure would be formed by placing two equilateral triangles base to base ? 81. Prove that any two sides of a triangle are together greater than the third side. Is the same proposition true of the angles of a triangle ? Give reasons for your answer. 32. Prove...

...plane rectilineal angle, and show how to bisect one, that is, to divide it into two equal angles. 2. Any two sides of a triangle are together greater than the third side. Is the same proposition true of the angles of a triangle? 3. What is a parallelogram? Prove that the...

...difference. Let two circumferences cut each other in the point A. Draw the radii CA, DA ; then, because any two sides of a triangle are together greater than the third side (Prop. VIII., B. I.), CD must be less than the sum of AD and AC. Also, DA must be less than the sum...

...line. What figure would be formed by placing two equilateral triangles base to base ? 31. Prove that any two sides of a triangle are together greater than the third side. Is the same proposition true of the angles of a triangle? Give reasons for your answer. 32. Prove geometrically...

...Petzval's arrangement ; for it appears that С 1 is greater than С m. The proof of this is easy enough. Any two sides of a triangle are, together, greater than the third, therefore 1 С and С A are together greater than A 1, and therefore than A m. Take away the common...

...given difference A D. The figure ABC therefore is the triangle required. PROP. 20.— THEOR. side. Any two sides of a triangle are together greater than the third CONSTRUCTION. — Pst. 2. A st. line may be lengthened out in a st. line. P. 3. From the greater line...

...Petzval's arrangement ; for it appears that CO is greater than Cm. The proof of this is easy enough. Any two sides of a triangle are, together, greater than the third, therefore OC and CA are together greater than AO, and therefore than Km. Take away the common part...

...the sides also which subtend, or are opposite to the equal angles, shall be equal to one another. 4. Any two sides of a triangle are together greater than the third side. The difference of any two sides of a triangle is always less than the third side. 5. The three interior...