Tiêu đề 41 Secondary Parts of a Triangle.pdf ✅

MSTC 41: SECONDARY PARTS OF A TRIANGLE 1. Lines 1.1. Median The median of a triangle is the line segment that joins a vertex and the midpoint of its opposite side. • It divides a triangle into two equal areas. • From Apollonius Theorem, b 2 + c 2 = 2 [ma 2 + ( a 2 ) 2 ] Similarly, the other medians produce: c 2 + a 2 = 2 [mb 2 + ( b 2 ) 2 ] a 2 + b 2 = 2 [mc 2 + ( c 2 ) 2 ] Apollonius Theorem is a special case of the Stewarts Theorem (Discussed in Other Theorems in Triangle) 1.2. Angle Bisector It is a line segment drawn from the vertex that bisects the interior angle of a triangle. • It divides a triangle into two similar triangles (Angle Bisector Theorem) • The length of the angle bisector:
[DERIVATION] Express the area of the triangle [ABC] = A1 + A2 1 2 bc sin A = 1 2 nac sin A 2 + 1 2 nab sin A 2 Solve for na na = bc sinA (b + c) sin A 2 na = bc (2 sin A 2 cos A 2 ) (b + c) sin A 2 na = 2bc cos A 2 b + c From cosine law cos A = b 2 + c 2 − a 2 2bc cos A 2 = √ 1 + cos A 2 cos A 2 = √ 1 + b 2 + c 2 − a 2 2bc 2 cos A 2 = √ (b + c + a)(b + c − a) 4bc Setting up s = a+b+c 2 as the semi-perimeter, cos A 2 = √ s(s − a) 2bc Substituting to the formula, na = 2bc√ s(s − a) 2bc b + c Simplifying gives the formula na = √2bcs(s − a) b + c Similarly, for the other angle bisectors, nb = √2cas(s − ab) c + a nc = √2abs(s − c) a + b

2. Points 2.1. Centroid It is the point of concurrency of the medians. • The three medians divide the triangle into six equal areas. • The centroid divides a median into a ratio of 2: 1. 2.2. Incenter It is the point of concurrency of the angle bisectors. The incenter is also the center of the circle inscribed in the triangle. The point is also equidistant from the sides.