Circumcenter, Incenter and Orthocenter; Lengths of angle bisector and median of a triangle. скачать в хорошем качестве

Circumcenter, Incenter and Orthocenter; Lengths of angle bisector and median of a triangle.

To prove that a triangle is dissected by its medians into six smaller triangles of equal area. To prove that the medians of a triangle divide one another in the ratio 2: 1; in other words, the medians of a triangle "trisect" one another. To prove each angle bisector of a triangle divides the opposite side into segments proportional in length to the adjacent sides. To prove that the internal bisectors of the three angles of a triangle are concurrent. The following problems are solved in the video with explanation 1. Prove that the circumcenter and orthocenter of an obtuse-angled triangle lie outside the triangle. 2. Find the ratio of the area of a given triangle to that of a triangle whose sides have the same lengths as the medians of the original triangle. 3. Prove that any triangle having two equal medians is isosceles. 4. Prove that any triangle having two equal altitudes is isosceles. 5. Use converse of Ceva's Theorems and the Theorem "Each angle bisector of a triangle divides the opposite side into segments proportional in length to the adjacent sides."1 to obtain another proof of Theorem "The internal bisectors of the three angles of a triangle are concurrent." 6. Find the length of the median A A' of the triangle ABC in terms of a, b, c. Hint: Use Stewart's theorem (Exercise 4 of Section 1.2). 7. Prove that the square of the length of the angle bisector AL is bc[1-{a/(b+c)}²] 8. Find the length of the internal bisector of the right angle in a triangle with sides 3, 4, 5. 9. Prove that the product of two sides of a triangle is equal to the product of the circumdiameter and the altitude on the third side. Law of sine Mathematics Olympiad Solution of Geometry Revisited by H. S. M. Coxeter & S. L. Greitzer. Geometry Revisited Advanced tropics in Geometry 1. Show that, for any triangle ABC, even if B or C is an obtuse angle, a = b cos C + c cos B. Use the Law of Sines to deduce the "addition formula" sin (B + C) = sin B cos C + sin C cos B. 2. In any triangle ABC, prove that, a (sin B - sin C) + b (sin C - sin A) + c (sin A - sin B) = 0. 3. In any triangle ABC, prove that ar(ABC) = abc/4R. 4. Let p and q be the radii of two circles through A, touching BC at B and C, respectively. Then prove that pq = R².    • MATHS OLYMPIAD, GEOMETRY