3. Prove that AD × DC + fc2 + fe2 = df2 + fe2, and that these, 4. 5. 6. B — EB2 + BD3, that AD X DC + EB2 = EB2 + BD3, Cor. If from a point without a circle two right lines, as AB, AC, be drawn cutting the circle, then Ab × AE AC X AF. The proof of this is merely that AB X AE and AC X AF are each = AD2, and each other. PROPOSITION XXXVII. Theorem. If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it; if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square of the line which meets it, the line which meets shall touch the circle. Steps of the Demonstration. 1. Prove that FED is a right, F that DE2 = AD × DC, and .. = db3, that (in AS DFB, dfe) ≤ def = ≤ DBF, 2. 3. 4. 5. that DBF is a right 2, 1. Prove that all the s at A, B, and c are rt. that S AMB + AKB = 2 rt. <s, and 2. DEF, B E PROPOSITION IV. Problem. To inscribe a circle in a given triangle. Steps of the Demonstration. 1. Prove that (in AS DBE, DBF) side DE side DF, by xxvi. I, 2. Prove that similarly DG = DF; and .. DE, DF, DG 3. that a each other, described with centre D, and distance either of these lines, will touch the sides AB, BC, CA, and .. that EFG is inscribed in ▲ ABC. Steps of the Demonstration. (In the construction of the figure, prove that the perpendiculars DF and EF will meet, as in F). base af, 1. Prove that (in AS ADF, BDF) base BF that similarly CF = AF; and .. af, bf, CF = each other, 2. 3. that a described with centre F, and distance either of these lines, will pass through the extremities of the other two, and be described about the ▲ ABC. PROPOSITION VI. Problem. To inscribe a B square E in a given circle. Steps of the Demonstration. 1. Prove that (in AS ABE, AED) base AB = base AD, that similarly BC, CD = BA, or AD; and .. 2. AB, BC, CD, DA = each other, that BAD is a right 4, that the figure ABCD is rectangular, PROPOSITION VII. G F B H K Problem. To describe a square D about a given circle. Steps of the Demonstration. 1. Prove that s at a are right s; and that simi larly s at B, C, and D are rights. F |