...describe the arc PM, which will be the perpendicular required. PROPOSITION xx. THEOREM. Every plane perpendicular to a radius at its extremity, is a tangent to the sphere. Let FAG be a plane perpendicular to the radius OA. Any point M in this plane being assumed,...

...circumference of 2d =£. rt _ R cos. 0 R cos. <£' • _cos. <p coS. 0' PROPOSITION m. Every plane perpendicular to a radius at its extremity is a tangent to the sphere in that point. Let ZXY be a plane perpendicular to the radius OZ. Then ZXY touches the sphere...

...the circumference in C, and must evidently cut it in a second point D. PROPOSITION XI— THEOREM. 26. A straight line perpendicular to a radius at its extremity is a tangent to the circle. Let AB be perpendicular to the radius OC at its extremity C; then, AB is a tangent to the circle at the...

...not the perpendicular from C to AB; therefore, AB is oblique to CD. 133. Proposition XIV.— Theorem. A straight line perpendicular to a radius at its extremity is a tangent to tlie circumference; and conversely, a tangent to a circumference is perpendicular to the radius drawn...

...shortest line from C to DE, and is, therefore, perpendicular to DE (I. 91). 28. Corollary. Conversely, a straight line perpendicular to a radius at its extremity is a tangent to the circumference (I. 39). THEOREM VIII. 29. The angle made by a tangent and a chord, intersecting at the...

...circumference in C, and must evidently cut it in a second point D. PROPOSITION XI.— THEOREM. 26. A straight line perpendicular to a radius at its extremity is a (angent to the circle. Let AB be perpendicular to the radius OC at its extremity C; then, AB is a tangent...

...shortest line from C to DE, and is, therefore, perpendicular to DE (I. 91). 28. Corollary. Conversely, a straight line perpendicular to a radius at its extremity is a tangent to the circumference (I. 39). THEOREM VIII. 29. The anrjle made by a tangent and a chord, intersecting at...

...§52 (a JL is the shortest distance from a point to a straight line). .-. much more is 0 К > OH. QED PROPOSITION IX. THEOREM. 186. A straight line perpendicular...straight line perpendicular to BA at A. We are to prove MO tangent to the circle. From B draw any other line to M 0, as B С Н. BН>BA, §52 (a JL measures...

...52 (a _L is the shortest distance from a point to a straight line). .-. much more is 0 K > 0 H. QED PROPOSITION IX. THEOREM. 186. A straight line perpendicular...tangent to the circle. From B draw any other line to M 0, as BC H. BH>BA, §52 (a _L measures the shortest distance from a point to a straight line). .'....

...circumference with OA as a radius passes through these points. ELEMENTS OF PLANE GEOMETRY. THEOREM IX. 194. A straight line perpendicular to a radius at its extremity is a tangent to the circle. Let AB be _L to the radius OP at P. To prove that AB is a tangent to the O at the point P. From the centre...