FIGURE 8.8: Two pairs of tangent circles Theorem 8.1.15 (Tangent Circles Theorem). If the circles ?1 = C(O1, r1) and ?2 = C(O2, r2) are tangent at P, then the centers O1 and O2 are distinct, and the three points O1, O2, and P are collinear. Furthermore, the circles share a common tangent line at P. Proof. Exercise 5. EXERCISES 8.1 1. Complete the proof of the Tangent Line Theorem (Theorem 8.1.7). 2. Prove that points on a tangent line lie outside the circle (Theorem 8.1.8). 3. Prove the Secant Line Theorem (Theorem 8.1.9). 4. Prove that some points on a secant line lie outside the circle and some lie inside (Theorem 8.1.10). 5. Prove the Tangent Circles Theorem (Theorem 8.1.15). 6. Let ? = C(O,r) be a circle, let l and m be two nonparallel lines that are tangent to ? at the points P and Q, and let A be the point of intersection of l and m. Prove the following (in neutral geometry). (a) O lies on the bisector of ?PAQ. (b) PA = QA. (c) If PQ intersects OA at R, then PR = QR. 7. This is an exercise in neutral geometry. (a) Let a and b be two numbers such that 0 < a < b. Prove that there exists a triangle ?ABC such that ?BCA is a right angle, BC = a, and AB = b. (b) Let ? be a circle and let P be a point that is outside ?. Prove that there exist two lines through P that are tangent to ?. 8. Use the Pythagorean Theorem to prove Elementary Circular Continuity (Theorem 8.1.11) in Euclidean geometry. Lines and Triangles in Neutral Geometry: In the preceding section, we proved that three distinct, collinear points cannot all lie on a circle. Now we ask whether or not three noncollinear points lie on a circle. The question is this: Given three noncollinear points, is there a circle that contains all three? The answer, at least in neutral geometry, is not necessarily. This may surprise you, since in high school geometry you probably remember that three noncollinear points always determine a circle. We will show that the assertion that three noncollinear points lie on a circle is equivalent to the Euclidean Parallel Postulate.
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Neutral SMSG Axioms 1. Given any two distinct points there is exactly one line that contains them. 2. Distance Postulate To every pair of distinct points there corresponds a unique positive number. This number is called the distance between the two points. 3. Ruler Postulate The points of a line can be placed in a correspondence with the real numbers such that: - To every point of the line there corresponds exactly one real number. - To every real number there corresponds exactly one point of the line. - The distance between two distinct points is the absolute value of the difference of the corresponding real numbers. 4. Ruler Placement Postulate Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive. 5. Every plane contains at least three non-collinear points. 6. If two points lie in a plane, then the line containing these points lies in the same plane. 7. Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane. 8. If two planes intersect, then that intersection is a line. 9. Plane Separation Postulate Given a line and a plane containing it, the points of the plane that do not lie on the line form two sets such that each of the sets is convex and if P is in one set and Q is in the other, then segment PQ intersects the line. 10. Space Separation Postulate. The points of space that do not lie in a given plane form two sets such that each of the sets is convex, and if P is in one set and Q is in the other, then segment PQ intersects the plane. 11. Angle Measurement Postulate To every angle there corresponds a real number between 0° and 180°. 12. Angle Construction Postulate. Let AB be a ray on the edge of the half-plane H. For every r between 0° and 180° there is exactly one ray AP, with P in H such that m∠PAB = r. 13. Angle Addition Postulate If D is a point in the interior of ∠BAC, then m∠BAC = m∠BAD + m∠DAC. 14. Supplement Postulate If two angles form a linear pair, then they are supplementary. 15. SAS Postulate Given a one-to-one correspondence between two triangles (or between a triangle and itself). If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence.
Sri K.
you will derive expressions for the radii of the circumscribed circle and the inscribed circle for $\triangle A B C .$ In these exercises, assume as given the following two results from geometry: i. The three angle bisectors of the angles of a triangle meet in a point. This point (labeled I in the figure) is the center of the inscribed circle. ii. The perpendicular bisectors of the sides of a triangle meet in a point. This point (labeled O in the figure) is the center of the circumscribed circle. The following figure shows the circumscribed circle for $\triangle A B C$. The point $O$ is the center of the circle, and $\mathscr{R}$ is the radius. In this exercise you will derive the formulas for the radius $\mathscr{R}$ of the circumscribed circle for $\triangle A B C:$ $$ \mathscr{R}=\frac{a}{2 \sin A}=\frac{b}{2 \sin B}=\frac{c}{2 \sin C} $$ (a) According to a theorem from geometry, the measure of $\angle A O C$ is twice the measure of $\angle B .$ What theorem is this? (State the theorem using complete sentences.) (b) Draw a perpendicular from $O$ to $A C$, meeting $\overline{A C}$ at $T$. Explain why $A T=T C=b / 2 .$ (As usual, $b$ denotes the length of the side opposite angle $B$.) Hint: Result (i) or (ii) above may be useful. (c) Explain why $\triangle A T O$ is congruent to $\triangle C T O$. (d) Use the results in parts (a) and (c) to show that $\angle C O T=\angle B$ (e) Use the result in part (d) to show that $\mathscr{R}=b /(2 \sin B)$ From this we can conclude (by the law of sines) that $$ \mathscr{R}=\frac{a}{2 \sin A}=\frac{b}{2 \sin B}=\frac{c}{2 \sin C} \text { as required } $$ (f) In the figure that we used for parts (a) through (e), the center of the circle falls within $\triangle A B C$. Draw a figure in which the center lies outside of $\triangle A B C$ and prove that the equation $\mathscr{R}=b /(2 \sin B)$ is true in this case, too. (g) Two triangles have the same numerical value for the ratio that appears in the law of sines (the length of a side to the sine of its opposite angle). What geometric property do the two triangles have in common?
Additional Topics in Trigonometry
The Law of Sines and the Law of Cosines
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Geometry A Common Core Curriculum
Geometry
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