Tiêu đề 43 Miscellaneous Theorems on Triangles.pdf ✅

MSTC 43: Miscellaneous Theorems on Triangles This discussion shows the different theorems that may be helpful when solving problems involving triangles. 1. Stewarts Theorem It states that for a triangle a 2m + c 2n = b(d 2 + mn) Use the formula to determine the length of a cevian. In this figure, the cevian is d. It is a line segment drawn from a vertex to an opposite side. [DERIVATION] From the cosine law on the left triangle, c 2 = m2 + d 2 − 2md cos θ From cosine law on the triangle to the right, a 2 = d 2 + n 2 − 2nd cos(180° − θ) a 2 = d 2 + n 2 + 2nd cos θ

2. Area Ratio This simple area-ratio theorem relates the ratio of the areas of two triangles with two of their sides. It also involves three collinear points. From the possible cases, [ABC] [ADC] = BP PD The two succeeding theorems were proven using this simple area theorem.
2.1. Ceva’s Theorem This theorem states that for concurrent cevians, AE EC ⋅ CF FB ⋅ BD DA = 1 This theorem is helpful to determine some of the division lengths of each side of the triangle. 2.2. Menelaus’ Theorem This theorem states that for two cevians, BD DA ⋅ AC CE ⋅ EG GB = 1 CE EA ⋅ AB BD ⋅ BG GE = 1