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PROP. XXVII. THEOREM.
The rectangle of the two diagonals of any quadrilateral, infcribed in a circle, is equal to the fum of the rectangles of its oppofite fides.
Let ABCD be any quadrilateral infcribed 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 alfo equal to the angle DBC, being angles in the fame fegment, whence the remaining angle AED is equal to the remaining angle ECD (I. 28. Cor.)
Since, therefore, the triangles ADE, BDC are equiangular, AD is to AE as BD is to BC (VI. 5.); and confequently the rectangle of AD, BC is equal to the rectangle of AE, BD (VI. 12.)
Again, the angle CDE being equal to the angle ADE (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, alfo equiangular; whence AB is to BD as EC is to DC (VI. 5.); and confequently the rectangle of AB, DC is equal to the rect angle of EC, BD (VI. 12).
And if, to thefe 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
Q. E. D.
1. The common fection 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 fection, is perpendicular to the other.
4. The inclination 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 fection, at right angles to that fection; one in one plane, and the other in the other.
6. Parallel planes, are fuch as being produced ever so far both ways will never meet.
7. A plane is faid to be extended by, or to pass through a right line, when every part of that line lies in the plane.
PROP. I. THEOREM.
The common fection of any two planes is a right line.
Let AB, CD be two planes, whose common fection is EF; then will EF be a right line.
For if not, let FGE be a right line, drawn in the plane
and FKE another right line, drawn in the plane CD. Then, fince the lines FGE, FKE are in different planes, they must fall wholly without each other.
But the line FGE, having the fame extremities with the line FKE, will coincide with it: whence they coincide and fall wholly without each other, at the fame time, which is abfurd.
The lines FGE, FKE cannot, therefore, be right lines; and confequently the line EF, which lies in each of the planes, must be a right line, as was to be fhewn.
SCHOLIUM. One part of a right line cannot be in a plane, and another part out of it. For fince the line can be produced in that plane, the part out of the plane, and the part produced would have different directions, which is abfurd.
PRO P. II. THEOREM.
Any three right lines which mutually interfect each other, are all in the fame plane.
Let AB, BC, CA be three right lines, which interfect each other in the points A, B, C; then will those lines be in the fame plane.
For let any plane AD pass through the points A, B, and be turned round that line, as an axis, till it pass through the point c.
Then, because the points A, C are in the plane AD, the whole line AC muft alfo be in it; or otherwife its parts would not lie in the fame direction.
And, because the points B, C are alfo in this plane, the whole line BC muft likewife be in it; for the fame reason.
But the line AB is in the plane AD, by hypothefis ; whence the three lines AB, BC, CA are all in the fame plane, as was to be fhewn.
COR. Any two right lines which interfect each other, are both in the fame plane; and through any three points a plane may be extended.