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4.2.6 Triangle Inequality and Median/Altitude Properties

Applying triangle inequality theorem and understanding properties of medians, altitudes, and angle bisectors.

定义

The Triangle Inequality Theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the sum of the lengths of any two sides must be greater than the length of the third side. Additionally, medians, altitudes, and angle bisectors are special line segments in a triangle with distinct geometric properties. A median connects a vertex to the midpoint of the opposite side and divides the triangle into two equal areas. An altitude is a perpendicular line segment from a vertex to the line containing the opposite side (or its extension). An angle bisector divides an angle into two congruent angles. The centroid (where medians intersect) divides each median in a 2:1 ratio from vertex to midpoint. The orthocenter is the intersection point of all three altitudes, and the incenter is where all three angle bisectors meet.

核心公式

  • \(a + b > c, \quad b + c > a, \quad a + c > b\)
  • \(|a - b| < c < a + b\)
  • \(m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}\)
  • \(\text{Centroid divides median in ratio } 2:1 \text{ from vertex}\)
  • \(\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{altitude}\)

易错点

  • ⚠️ Confusing the triangle inequality with the reverse inequality—students often write \(a + b < c\) instead of \(a + b > c\), or forget that the sum must be strictly greater than (not equal to) the third side
  • ⚠️ Incorrectly applying the centroid ratio—students may reverse the 2:1 ratio or measure from the wrong endpoint, forgetting that the centroid divides each median in a 2:1 ratio from the vertex to the midpoint
  • ⚠️ Mixing up altitudes with medians—students sometimes assume an altitude bisects the opposite side (which is only true for isosceles/equilateral triangles) or that a median is perpendicular to the opposite side
  • ⚠️ Misidentifying the orthocenter location—students may place the orthocenter inside the triangle for all cases, forgetting that it lies outside (obtuse triangle) or on the triangle (right triangle)