he equal to the rectangle of AD, DB, together with the fquare of cD. For, about the triangle ABC, describe the circle AEC (IV.5.), cutting cd, produced, in E; and join EB. Then, because the angle acp is equal to the angle ECB (by Hyp.), and the angle CAD to the angle CEB (III. 15.), the remaining angle ADC will be equal to the remaining angle CBE (1. 28. Cor.) The triangles CAD, ces being, therefore, equiangular, ÇA will be to CD as ce to CB (VI. 5.); and consequently the rectangle of ÇĀ, ce is equal to the rectangle of çe, CD (VI. 12.) But the rectangle of ce, cd is equal to the rectangle of ED, DC, together with the square of CD (II. 19.); whence the rectangle of CA, CB is also equal to the rectangle of ED, DC, together with the square of cD. And since the rectangle of ED, DC is equal to the rect, angle of AD, DB (III. 27.), the rectangle of AC, CB is also equal to the rectangle of AD, DB, together with the square of CD. PROP. XXVI. THEOREM. The rectangle of the two sides of any tri. angle, is equal to the rectangle of the perpendicular, drawn from the vertical angle to the base, and the diameter of the circumscribing circle. Let ABC be a triangle, having cd perpendicular to AB; then will the rectangle of Ac, co be equal to the rectangle of cD and the diameter of the circumscribing circle. For, about the triangle ABC, describe the circle AEC (IV.5.); in which draw the diameter ce; and join EB. Then, since the angle cad is equal to the angle CEB (III. 15.) and the angle adc to the angle Eec, being each of them right angles (Const. and III. 16.), the remaining angle ACD will be equal to the remaining angle ecb (I. 28. Cor.) The triangles ACD, ECB are, therefore, equiangular; whence Ac is to cd as ce is to CB (VI. 5.); and consequently the rectangle of AC, CB is equal to the rectangle of CD, CE (VI. 12.) Q. E. D. SCHOLIUM. When abc is an obtuse angle, the perpendicular cp falls without the circle ; but the fame demonftration will hold. PRO P. XXVII. THEOREM. The rectangle of the two diagonals of any quadrilateral, inscribed in a circle, is equal to the sum of the rectangles of its opposite sides. A B В Let ABCD be any quadrilateral inscribed in a circle, of which the diagonals are AC, BD; then will the rectangle of AC, BD be equal to the rectangles of AB, DC and AD, BC. For make the angle cde equal to the angle ADB (I. 20.); then, if to each of these angles, there be added the common angle edB, the angle ADE will be equal to the angle CDB. The angle dae is also equal to the angle DBC, being angles in the fame segment, whence the remaining angle AED is equal to the remaining angle BCD (I. 28. Cor.) Since, therefore, the triangles ADE, BDC are equiangular, AD is to AE as bp is to BC (VI. 5.); and consequently the rectangle of AD, BC is equal to the rectangle of AE, BD (VI. 12.) Again, the angle CDE being equal to the angle ADB (by Conft.), and the angle EcD to the angle ABD (III. 15.), the remaining angle CED will be equal to the remaining angle BAD (I. 28. Cor.) The triangles CED, ADB are, therefore, also equiangular ; whence AB is to BD as ec is to DC (VI. 5.); and consequently the rectangle of AB, DC is equal to the rectangle of ec, BD (VI. 12). And if, to these equals, there be added the former, the rectangle of AB, DC together with the rectangle of AD, BC will be equal to the rectangle of EC, BD together with the rectangle of AE, BD. But the rectangles of AE, BD, and EC, BD are equal to the rectangle of AC, BD (II.8.); whence the rectangle of AC, BD is also equal to the rectangles of AB, DC and AD, BC, Q. E. D. BOOK. VII. DEFINITIONS. I. The common section of two planes, is the line in which they meet, or cut each other. 2. A right line is perpendicular to a plane, when it is perpendicular to every right line which meets it in that plane. 3. A plane is perpendicular to a plane, when every right line in the one, which is perpendicular to their common section, is perpendicular to the other. 4. The inelination of a right line to a plane, is the angle it makes with another line, drawn from the point of section, to that point in the plane, which is cut by a perpendicular falling from any part of the former. 5. The inclination of a plane to a plane, is the angle contained by two right lines, drawn from any point in the common section, at right angles to that section ; one in one plane, and the other in the other. 6. Parallel planes, are such as being produced ever so far both ways will never meet. 7. A plane is said to be extended by, or to pass through a right line, when every part of that line lies in the plane. |