PROPOSITION X. THEOREM 313. A tangent to a circle is perpendicular to the radius drawn to the point of tangency. Given line AB, tangent to circle O at T, and OT, a radius drawn to the point of tangency. To prove ABOT. ARGUMENT 1. Let M be any point on AB other than T; then M is outside the circumference. REASONS 1. § 286. 6. .. OT is the shortest line that can be drawn from 0 to AB. 7. .. OT LAB; i.e. AB ↓ OT. Q.E.D. 7. § 165. 314. Cor. I. (Converse of Prop. X). A straight line perpendicular to a radius at its outer extremity is tangent to the circle. HINT. Prove by the indirect method. In the figure for Prop. X, suppose that AB is not tangent to circle O at point T; then draw CD through T, tangent to circle O. Apply § 63. 315. Cor. II. A perpendicular to a tangent at the point of tangency passes through the center of the circle. 316. Cor. III. A line drawn from the center of a circle perpendicular to a tangent passes through the point of tangency. 317. Def. A polygon is circumscribed about a circle if each side of the polygon is tangent to the circle. In the same figure the circle is said to be inscribed in the polygon. Ex. 437. The perpendiculars to the sides of a circumscribed polygon at their points of tangency pass through a common point. Ex. 438. The line drawn from any vertex of a circumscribed polygon to the center of the circle bisects the angle at that vertex and also the angle between radii drawn to the adjacent points of tangency. Ex. 439. If two tangents are drawn from a point to a circle, the bisector of the angle between them passes through the center of the circle. Ex. 440. The bisectors of the angles of a circumscribed quadrilateral pass through a common point. Ex. 441. Tangents to a circle at the extremities of a diameter are parallel. PROPOSITION XI. PROBLEM 318. To construct a tangent to a circle at any given point in the circumference. The construction, proof, and discussion are left as an exercise for the student. (See § 314.) Ex. 442. Construct a quadrilateral which shall be circumscribed about a circle. What kinds of quadrilaterals are circumscriptible? Ex. 443. Construct a parallelogram which shall be inscribed in a circle. What kinds of parallelograms are inscriptible? Ex. 444. Construct a line which shall be tangent to a given circle and parallel to a given line. Ex. 445. Construct a line which shall be tangent to a given circle and perpendicular to a given line. 319. Def. The length of a tangent is the length of the segment included between the point of tangency and the point from which the tangent is drawn; as TP in the following figure. PROPOSITION XII. THEOREM 320. If two tangents are drawn from any given point to a circle, these tangents are equal. T Given PT and PS, two tangents from point P to circle 0. The proof is left as an exercise for the student. Ex. 446. The sum of two opposite sides of a circumscribed quadrilateral is equal to the sum of the other two sides. Ex. 447. The median of a circumscribed trapezoid is one fourth the perimeter of the trapezoid. Ex. 448. A parallelogram circumscribed about a circle is either a rhombus or a square. Ex. 449. The hypotenuse of a right triangle circumscribed about a circle is equal to the sum of the other two sides minus a diameter of the circle. Ex. 450. If a circle is inscribed in any triangle, and if three triangles are cut from the given triangle by drawing tangents to the circle, then the sum of the perimeters of the three triangles will equal the perimeter of the given triangle. 321. To inscribe a circle in a given triangle. 1. Construct AE and CD, bisecting CAB and BCA, respectively. §127. 2. AE and CD will intersect at some point as 0. 3. From o draw OF AC. § 149. § 194. 4. With O as center and OF as radius construct circle FGH. 5. Circle FGH is inscribed in ▲ ABC. II. The proof and discussion are left for the student. B 322. Def. A circle which is tangent to one side of a triangle and to the other two sides prolonged is said to be escribed to the triangle. Ex. 451. Problem. To escribe a circle to a given triangle. Ex. 452. (a) Prove that if the lines that bi sect three angles of a quadrilateral meet at a com A mon point P, then the line that bisects the remaining angle of the quadrilateral passes through P. (b) Tell why a circle can be inscribed in this particular quadrilateral. Ex. 453. In triangle ABC, draw XY parallel to BC so that XY+ BC= BX + CY. Ex. 454. Inscribe a circle in a given rhombus. PROPOSITION XIV. PROBLEM 323. To circumscribe a circle about a given triangle. M B Given ABC. To circumscribe a circle about ▲ ABC. The construction, proof, and discussion are left as an exercise for the student. 324. Cor. Three points not in the same straight line determine a circle. Ex. 455. Discuss the position of the center of a circle circumscribed about an acute triangle; a right triangle; an obtuse triangle. Ex. 456. Circumscribe a circle about an isosceles trapezoid. Ex. 457. Given the base of an isosceles triangle and the radius of the circumscribed circle, to construct the triangle. Ex. 458. The inscribed and circumscribed circles of an equilateral triangle are concentric. Ex. 459. If upon the sides of any triangle equilateral triangles are drawn, and circles circumscribed about the three triangles, these circles will intersect at a common point. Ex. 460. The two segments of a secant which are between two concentric circumferences are equal. Ex. 461. The perpendicular bisectors of the sides of an inscribed quadrilateral pass through a common point. Ex. 462. The bisector of an arc of a circle is determined by the center of the circle and another point equidistant from the extremities of the chord of the arc. Ex. 463. If two chords of a circle are equal, the lines which connect their mid-points with the center of the circle are equal. |