be equal to the rectangle of AD, DB, together with the fquare of CD.

For, about the triangle ABC, defcribe the circle ABC (IV. 5.), cutting CD, produced, in E; and join EB.

Then, because the angle ACD 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 (I. 28. Cor.)

The triangles CAD, CEB being, therefore, equiangular, CA will be to CD as CE to CB (VI. 5.); and confequently the rectangle of CA, CB is equal to the rectangle of cɛ, CD (VI. 12.)

But the rectangle of CE, CD is equal to the rectangle of ED, DC, together with the fquare of CD (II. 10.); whence the rectangle of CA, CB is also equal to the rectangle of ED, DC, together with the fquare of CD.

And since the rectangle of ED, angle of AD, DB (III. 27.), the also equal to the rectangle of AD, fquare of CD.

DC is equal to the rectrectangle of AC, CB is DB, together with the

Q. E.D.

PRO P. XXVI. THEOREM.
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The rectangle of the two fides of any triangle, is equal to the rectangle of the perpendicular, drawn from the vertical angle to the base, and the diameter of the circum fcribing circle.

Let ABC be a triangle, having CD perpendicular to AB; then will the rectangle of AC, CB be equal to the rectangle of CD and the diameter of the circumfcribing circle.

For, about the triangle ABC, defcribe the circle AEC (IV. 5.); in which draw the diameter CE; and join EB.

Then, fince the angle CAD is equal to the angle CEB (III. 15.) and the angle ADC to the angle EBC, being each of them right angles (Conft. 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 confequently the rectangle of AC, CB is equal to the rectangle of CD, CE (VI. 12.) Q.E. D. SCHOLIUM. When ABC is an obtufe angle, the perpendicular CD falls without the circle; but the fame demonftration will hold.

PROP.

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 BCD (I. 28. Cor.)

Since, therefore, the triangles ADE, BDC are equiangular, AD is to AE as BD is to BC (VI. 5.); and confe quently 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 (1. 28. Cor.)

The triangles CED, ADB are, therefore, alfo 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. (VÍ. 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 AD, BC.

BOOK VII.

DEFINITIONS.

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-section, 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 fection, 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 fo far both ways will never meet.

7. A plane is faid to be extended by, or to pafs through a right line, when every part of that line lies in the plane.